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c.e.c essay Our goal is to help you find the angle in eur your essay that makes you shine. Kkk Essays Papers! We work with you first to communicate your story as powerfully as possible – to craft your ideas around a compelling theme and reveal your passions and your personality. We then help you to polish your expression, grammar, and use of language to thesis eur, present the outline best of eur, your writing. Essay! If you are interested in the actual editing process please take a look at thesis how we edit. This is writing, a fine example of thesis, how to doctorate, show the reader where you have come from. I remember the bright lights. Thesis! That’s the earliest memory I can faithfully recollect, the first point in my timeline. Business Life Cycle! I remember the warmth of the moment, the eur faint glimmer of neon on the streets below the apartment. Only beginning with this moment can I explore the whale essays full extent of my memories, the thesis events of childhood that shaped my life while I lived with my grandparents in a city in China that my ancestors had resided in for ten generations.

I have other bits and pieces of, patchy recollections. Thesis Eur! Together, they make up the hours and days of my childhood. Ib History Essay Outline! It is a time that I remember fondly today, although I’m sure the thesis gaps in my memories have often been clouded by half truths, pieced together from essays random photographs and my own impressions. These memories are my constant reminder of what has become my own basic philosophy on eur, life. On Water! I remember a trip I made to the park, riding an old, but still brightly colored merry-go-round, vibrant with hues of red, blue, and yellow. Thesis Eur! I see my grandfather, helping set the large wheel into research motion as the colors swirled gracefully. While I was in China, my parents studied diligently in Lawrence, Kansas.

I know this because after coming to thesis eur, America, I grew up around high-performance liquid chromatography, mass spectroscopy, and doctorate programs, reaction mechanisms; I remember minute details from the thesis eur amount of whale essays, time I spent in thesis eur the cramped quarters of the lab. On that merry-go-round and life, even just five years ago, I never saw the toll placed on thesis, both sides of reports college, my family. I remember mealtimes with my cousin in eur the siheyuan, the courtyard, by the house we used to live in coursework programs Huaiyin. Today, I can imagine the glow of the sun and the scent of the greasy, fried delicacies of the vendors just an thesis alley away. I hear the bustle of the cars, but mainly pedestrians and bicycles going to work in the city. But most of all, I feel the care of my grandmother and my aunt, feeding both my cousin and me while we watched the birds fly through the hazy sky. Thesis Eur! Today, the setting is outline, different – a rural New England town, surrounded by the vivid colors of falling leaves.

My childhood has never stayed far behind. Thesis! I now know about the sacrifices that my parents made on essays on water, my behalf. They never heard my first words, saw my first steps, nor did they share the first bits of my childhood. But in eur the end, all the sacrifices were worth the whale essays effort, something I note with more insight every year: I can now see a chain of goodwill, one that has made me see the benevolence in people. In coming to thesis eur, America, I have one memory that sticks out among the rest. Crisis! Flying into eur a city, one whose name has become lost to essays in india, me, I once again saw the bright lights. Eur! They were the lights of a new country and a new life. I didn’t know why this memory always stood out at the time, but today, I do. I’ve lived two lives – the doctorate programs first began with the lights and an apartment, the eur second with the life cycle lights and eur, an airplane.! Today, I bridge the gap between the two, and I see that I am fortunate to thesis, have two lives and writing book college, people who care in each of.

This was written by a student applying to a business program. Thesis! T he current financial recession irks me. At the heart of the essay crisis that has gripped my community, the nation and eur, the world is dishonesty, avarice and, uncertainty. It is more than upsetting to recognize that unsavory business practices have hurt so many people across the globe. Not only thesis, does it undermine the basis on which business should be conducted, but it also goes against my own ethical code.

I have built a successful company from scratch without the thesis eur assistance of kkk essays research, mom, dad, or uncle based on thesis eur, fair and writing reports, honest business practices. I take great pride in the report cards I’ve received from the people with whom I do business. Thesis Eur! Over the ib history essay outline last four years, I have received a perfect “feedback score” from all of eur, my eBay customers. By perfect, I mean that every item I have sold arrived on on water, time, shipping charges were deemed fair, the items were properly portrayed, and thesis, buyer-seller communication was excellent. I don’t claim that these practices are ingenious or innovative, but in my field of writing book, work the abundance of eur, chicanery is unsettling. It was four years ago that I re-acquainted myself with my old elementary school love, baseball cards. I spent an business cycle inordinate amount of thesis eur, time researching and buying everything pertaining to baseball cards. Essays Crisis! Some would consider instant messaging fellow card traders in thesis Japan at 3 a.m.! (our time) about thesis eur sell-values a cause for concern. Kkk Essays! I didn’t; my alarm was set for 2:55 a.m. Thesis Eur! and by 3:15 a.m. the transaction was completed and on water in india, I was back to sleep. It wasn’t too long before buyers and eur, sellers in both the Japanese and American markets started to whale essays, rely on thesis eur, my company. My office was my bedroom, my research center was the research family laptop, and eur, my distributor was the United States Postal Service.

While most of business essay, my friends were writing on thesis eur, each other’s Facebook Wall, I was on researching the price of life cycle essay, a 1955 Topps Sandy Koufax card. Thesis! I was already aware of doctorate programs, my obsessive personality, so I did not see any problem in thesis the fact that I could list the value of essays on water crisis in india, any card from the 1940s. Picasso had his blue period and his rose period, I was in my baseball cards period. Thesis Eur! The baseball card world is home to reports college, a plethora of fascinating opportunities and colorful people (to say the thesis least), but it is unfortunately overrun by, crooks and eur, cheats. Writing College! Whether it’s lying about the value and rarity of a card or misleading a prospective buyer about its condition, ubiquitous thieves are at both card shows and thesis, online auctions. Seeing middle school students lose as much as $500 to these online con artists made me realize that I wanted to doctorate, be the Martin Arrowsmith of the baseball card world. I did not turn a $1,500 investment into $10,000 by simply employing savvy business principles. Thesis Eur! Ethical business conduct played the primary role that led to a 567% growth in thesis eur my business. Buyers have flocked to research, me because they know I am an honest broker.

I have formed valuable friendships at thesis eur card shows and doctorate, online. Colleagues know they can count on me and eur, they are willing to crisis in india, return the favor when needed. The year my company began to flourish was my junior year. It was the same year that I finally started to thesis eur, tap into my true potential as a student. Was it a mere coincidence? I had found a passion that reinforced the kkk essays value of honesty; a passion that motivated and focused me, and it translated to eur, all aspects of my life. This was written by doctorate programs, a student who was accepted to Wesleyan University. Eur! B ringing home decorated pumpkins and the death of king arthur essay, gourds to your loving mom may seem a bit elementary for eur, a high school senior to arthur, do. If a high school bully sighted a student carrying a painted muskmelon with the words I love you Mom on it, he might open the door to the room where Teasing 101 is taught. I don’t run into that problem.

It’s not that we don’t have bullies in our school, but it helps that I’m very active in the weight room, and that I’m the eur captain of crisis in india, our football team. I’m also president of our school’s crafts club. I’m just as comfortable knitting a hat as I am making a tackle. Making a tackle is like making a get well card; you have to thesis, have all the right pieces. For the tackle you have to have feet placed properly, hips lowered, hands cocked precisely, and writing book college, plenty of heart. Eur! For the card you have to kkk essays papers, have paper, colored ink, stamps and plenty of thesis, heart.

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I know I have shaped their lives in some way; I have spent countless hours teaching them how to writing book, read, write and thesis, study Hebrew. Doctorate! I have learned how to thesis eur, run groups smoothly. I once heard a coach ask, Is the juice worth the squeeze? While the the death arthur essay whole team laughed at the time because it sounded so funny, I knew just what he meant. I often come back to that somewhat convoluted saying now that high school is coming to an end. Thesis Eur! Let me tell you, the juice has never tasted better. I enjoy making my mother feel good; that’s why I bring home corny little trinkets from Crafts Club that express my love. I enjoy the feeling I get making cards for whale essays, people who could use a lift. Thesis Eur! I enjoy being the fire in arthur my football team’s belly. Eur! But it’s not really about whale essays football or crafts club. Thesis! It’s about the joy these activities bring me and kkk essays research papers, the pride I gain from helping other people whenever I can – be it a new member on the squad trying to thesis eur, learn the plays, my “little brother,” or the students in my Hebrew school class.

It is how I purl every little aspect of my life into kkk essays a hat that I can wear and thesis, know that no one else is wearing the same one.

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Sample Of Nurse Student Clinical Self Reflection Paper Essays and Research Papers. One of characteristics of clinical leader: be supportive to new nurses Introduction In health care, the leader role in medical . Thesis Eur! units has evolved from solely medical to more managerial as well (Maddux, Maddux, amp;Hakim, 2008). It has been noted that leadership styles are important in doctorate transforming, creating meaning, and producing desirable employee outcomes; thus they can benefit organizational performance and even survival, especially for thesis eur new employee (McNeese-Smith,1995). Employee behaviors such. Clinical trial , Emotion , Health care 1212 Words | 4 Pages. Clinical Practices Tammy Turner ABA RN, BSN Hsn 552 November 4, 2013 Susan Dolinar PhD, RN, CNE Clinical Practices . For the needs of the patients, families, and the health care consumers, and writing book college, health care practice current assessment practices should be maintained by thesis using evidence –based practices. Health care professionals are expected to demonstrate competent practice within the society. Determining sufficient of clinical assessments and the evaluation of the nursing students has. Assessment , Bachelor of business essay Science in Nursing , Evaluation 1740 Words | 6 Pages. Reflection: Nursing and Professional Student Nurse. incident took place at a local Hospital during my thirty weeks placement.

In order to help me with my reflection I have chosen Gibbs (1988), . as the model to thesis, help guide my reflective process. Business Life Essay! This is an iterative model with six stopping points, using these stopping points as headings; I will be able to reflect fully on the incident. So, what is reflection ? According to Reid (1993), “ reflection is a process of reviewing an experience of practice in order to describe, analyse, evaluate and so inform. Diploma in Nursing , Florence Nightingale , Nurse 2065 Words | 7 Pages. The Learning Experience of a student nurse , a critical analysis. Thesis Eur! Prepared By: Azores, Mae Ann A. BSN 3D2-6 Submitted . To: Mr. Dennis Ubenia Eng5A- Instructor My learning throughout the first year has been helped by business life cycle essay an unerring optimism in the value of eur nursing, and business cycle essay, an appreciation that each and every daily interaction augments my experience. This enthusiasm, however, has caused an thesis eur, inhibitory effect on my self -directed researching, and created conflict in some placement areas. While. Bachelor of Science in Nursing , Florence Nightingale , Nurse 2525 Words | 7 Pages. verbal so that the writing message could be understood and do not break the thesis eur nurse -patient communication.

In my opinion, I evaluated that it does not a . Whale Essays! matter whether it was a patient-centred communication or task-centred communicationbecause both communication mentioned by thesis eur McCabe (2004) actually doesinvolves communication to life cycle, the patients. So it was not a problem to argue whichtype of eur communication involves in whale essays my conversation with my patient. After analysed the situation, I could conclude that I was be able. Communication , Health care , Health care provider 1294 Words | 7 Pages. Running Head: SAMPLE APA PAPER FOR STUDENTS RUNNING HEAD: Yes, it will say, “Running head: . . . .”, but only on . the first page. Use an abbreviated form of the title (not exceeding 50 characters, which will include spaces). Eur! The abbreviated title is in capital letters. This will appear in the death of king arthur essay the top left corner of every page PAGE NUMBERS: A page number will appear on every page in the top right corner of every page. Eur! 1 All elements in the dotted boxes are not visible on whale essays your final paper . Essay title. American Psychological Association , APA style , Bibliography 1615 Words | 6 Pages.

NAME MASENG THABO lethata STUDENT ID: 200901924 BNS311 (internship) ward: p.s.w . Third CLINCAL REFLECTION ( week 28 May-01 june 2012) Learning objectives: 1) Function collaboratively as members of staff 2) Serving as client advocate 3) Developing leadership and managerial skills Activities done to achieve the thesis objectives Functioning collaboratively as members of staff In order o achieve this objective I carried out. English-language films , Following , Hospital 581 Words | 3 Pages. Critical Reflection on One Nurse-Family Interaction. Introduction This paper describes a significant learning family interaction that occurred when I was working in a community placement as a . Book Reports! student nurse . I would be reflecting on that family interaction and use the “One Question Question” approach and thesis eur, analyse how this would contribute to your learning needs and whale essays, nursing practice. Critical or significant interactions can be defined as situations that make an impression or have a special meaning to an individual. The analysis of such incidents, a. Diabetes mellitus , Family , Family therapy 2306 Words | 6 Pages. Clinical practice is of outmost significance because this is the site where students can turn theory into practice, . students can interact with patients and eur, families, and face the real world of medicine. It is significant and useful to reflect upon assessment practices to evaluate students in clinical setting. For this purpose, this paper analyzes three articles relevant to the clinical practices by the nursing students , and book college, the role they should play outside the classroom, and who should evaluate. Best practice , Clinical nurse specialist , Education 1700 Words | 5 Pages.

Self Determination in the Elderly. Self -Determination in the Elderly 1 Nursing Students ’ Perceptions of Self -Determination in . Elderly People Maritta Valimaki, Helena Haapsaari, Jouko Katajisto and Ritta Suhonen Research and Design Professor Pollack April 10, 2007 Nursing Students ’ Perceptions of Self -Determination in Elderly People: A Critique ABSTRACT Self -Determination in the Elderly 2 This. Death , Elderly care , Geriatrics 2314 Words | 7 Pages. Introduction I am a registered nurse of twenty years of thesis experience. Business Life! After graduated from thesis, my three-year training in a nursing school, I had . On Water In India! been worked in an intensive care unit (ICU) and a pulmonary unit of a public hospital.

Now I am working in a non-government organization serving the thesis eur mentally disabled. No matter which specialty I am working in or how senior I am, I found learning being crucial for professional competence, job satisfaction and personal growth. In this paper , I would like to whale essays, reflect critically. Education , Educational psychology , Learning 1865 Words | 7 Pages. n Reflection Paper May 3, 2012 Theo 104 by Natasha S Morris . Introduction This course has taught me what it really means to thesis, be a Christian. I must admit I was not born into life cycle a Christian home, as I got older I started to learn about God and thesis, Jesus Christ. Essays On Water Crisis In India! My mother did teach me right from wrong but going to church and learning of God was not a priority in my home. Taking this course has taught me.

Bible , Christianity , God in Christianity 1797 Words | 5 Pages. B01BUIS213 Sample Paper For Students. SAMPLE FINAL EXAMINATION, Trimester 3, 2013 Unit Name: Business Information System Unit Code: B01BUIS213 To be completed by the . student : (Surname): (Given name) Name ID Number Course Title Signature Date This Question paper consists two parts (Part A, B, and thesis eur, C) Total time allowed : 2 hours 10 minutes (10 minutes perusal) All questions should be attempted. This is business cycle, a closed book examination. NO MATERIAL IS ALLOWED. Thesis Eur! Hardcopy dictionaries will be provided The questions may be answered in any. A Closed Book , Case study , Computer 630 Words | 3 Pages. ? SAMPLE Reflective Writing Assignments: COURSE: Going Local: Service-Learning to Link Entrepreneurship, Policy and Science (ASCI 297 and . CDAE/NFS/PPS 195/295) INSTRUCTOR: Richard Schramm Written assignments are of coursework two types: Reflections and thesis eur, Project Reports.

Reflections : An essential element for student learning in service-learning courses is written and essay, oral reflections on the field study experience, as well as on other elements of the course. To have an experience isn’t enough to ensure. Critical thinking , Literature , Paper 1243 Words | 4 Pages. ? Self - Reflection Paper Phelicia M. Williams Human Development/EDFN 990 November 22, 2013 Michael P. . McCreery Ph.D. Self - Reflection Paper When I think about my own biases, perceptions, attitudes, and beliefs I use to holds about thesis eur different culturally ethics groups, I definitely have to rethink and on water, correct myself and eur, not stereotype individuals on cycle essay my thoughts anymore. I always knew mankind was magnificent because we were created and GOD eyes but this course just emphasized just how important. Big Five personality traits , Neuroticism , Personality psychology 2024 Words | 8 Pages. ? Clinical Reflection Assignment 1 Brittany Dely University of Detroit Mercy- Henry Ford Macomb . Hospital In nursing school, we as students learn that critical thinking is an essential component in thesis providing quality nursing care.

According to Bittner and Tobin critical thinking is defined as being “influenced by knowledge and experience, using strategies such as reflective thinking as a part of learning to identify the issues and whale essays, opportunities. Anticonvulsant , Critical thinking , Florence Nightingale 826 Words | 6 Pages. Self Assessment in the Clinical Environment. INTRODUCTION This essay will discuss the advantages and disadvantages of eur student self assessment in clinical . Business Life Essay! environments and the importance of health professionals utilizing self assessment throughout their careers. Thesis Eur! Self assessment is the technique by which a student or professional rates and judges his or her own work thereby motivating him or her to work on eur improving any aspects of their work they feel may be lacking.

It is an assessment technique employed in a diverse range of writing book reports college settings worldwide. Dental hygienist , Education , Educational psychology 1783 Words | 5 Pages. Psychology Paper and thesis, Self Reflection. Self - Reflection Related to Influential Psychology Theories When psychologists were first acknowledged they began studying . Whale Essays! behaviors and biological processes. There are numerous recognized psychologists that have contributed to the seven major perspectives of today’s modern psychology. These theories include: Evolutionary theory, sociocultural theory, biopsychological theory, psychoanalytic theory, cognitivism, humanistic theory and eur, behaviorism. The three theories that I would like to describe. Behaviorism , Classical conditioning , Ivan Pavlov 1973 Words | 5 Pages. Mental Health Self Reflection Assignment 1. ? Mental Health Self Reflection Assignment December 1, 2014 Introduction Although theoretical preparation clearly is . important, there is crisis in india, no substitute for thesis the learning that takes place during clinical experiences. Clinical experience is whale essays, acknowledged as being the core of eur nursing education.

As nursing student without previews experience in mental health, I was reflecting the same attitude held by general public that mentally ill patient are dangerous, prone to coursework, violence, unpredictable, and thesis eur, to. Mental disorder , Nurse , Nursing 1178 Words | 6 Pages. INTENSIVE NURSING PRACTICUM 2nd Semester School Year 2011 - 2012 Clinical Teaching Plan Cagayan Valley Medical Center (Surgery Ward) . Prepared by: ____________________________________ Noted by: _________________________________________ Approved By: ______________________________________________ CLINICAL TEACHING PLAN DESCRIPTION OF LEARNERS * an whale essays, assessment tool used for identifying an thesis, individual student's strengths and needs in various content and developmental areas. Florence Nightingale , Health , Health care 1808 Words | 7 Pages. Reflection on a clinical Skill This essay will discuss a clinical skill in which I have become competent in . practicing as a student nurse . I will use a reflective model to discuss how I have achieved the necessary level of competence in my nurse training programme. The reflective model I have chosen to use is Gibbs model (Gibbs 1988). Gibbs model of reflection incorporates the following: description, feelings, evaluation, analysis, conclusion and an action plan (Gibbs 1988).

The model will. Evidence-based medicine , Gluteal muscles , Gluteus maximus muscle 1407 Words | 4 Pages. Reflection of life Clinical Observation Experience. Reflection on Temperature, Pulse, Respiration and Blood pressure I am a nursing student from Australian Catholic University. . This role play is about the vital signs. Thesis Eur! Now, I am going to whale essays, write a reflective write about the video of my role play of TPR which is a part of assignment and done by me and my other two friends. Firstly, in my video when I play a nurse role, first of all I asked about the patient’s condition such as how are you today. I also make the eye contact while taking to patient. Blood pressure , Communication , Nonverbal communication 710 Words | 3 Pages. Outline Your Developmental Trajectory as a Student Nurse, from Novice to Competent Practitione. developmental trajectory as a student nurse , from novice to competent practitioner.

This assignment objectively looks at my . Eur! developmental trajectory with growth in adult nursing from the novice level, advanced beginner and through to business life cycle, the competent practitioner level. During the thesis eur duration of my three years of nursing training I have come to realise the obligation on myself as a student and future staff nurse to build the whale essays theory and skills base expected of thesis eur qualified nurse . I shall look at cycle essay, how. Florence Nightingale , Nurse , Nurse education 1975 Words | 7 Pages. A Study of Nurse Caring Megan McMahon University of Illinois at Chicago A Study of Nurse Caring Although there are many . common themes in thesis eur both nursing practice and nursing theory, the concept of caring in of king nursing may be the thesis eur most central theme of in india them all. Thesis Eur! In nursing, the one word ‘caring’ can both refer to the verb meaning ‘the action of carrying out the on water crisis interventions and needs of a patient’, or the emotion or feeling of empathy and/or the eur altruism that puts the patient’s best interests first. Family centered care , Florence Nightingale , Nurse 1375 Words | 2 Pages. Learning Environment in Clinical Practice.

Clinical practice in hospital plays an important part of learning environment for nursing curriculum. Mc Cabe (1985) thinks that . clinical learning experience provides students for knowledge and socializing in professional education. Thesis! ‘Furthermore, the professional socialization of nursing students occurs largely in the practice setting’ (Windsor 1987.Lee French 1997). I am a registered staff nurse working in emergency department with experience almost for nine years. Before this I’m one of the. Education , Educational psychology , Learning 2084 Words | 7 Pages. contribution of reflective practice for clinical nursing. Essays Crisis! Reflection has been defined as a way for individuals to “capture their . experience, think about thesis eur it, mull it over and evaluate” (Boud et al 1985: 19) Argyris and Schon (1974) suggest that practitioners often practice at less than effective levels because they follow routine. Life Cycle Essay! Johns (1995) implies that action can be taken through reflection to increase effectiveness in practice as reflection provides opportunities for self development as professionals and. Donald Schon , John Wiley Sons , Nurse 1549 Words | 5 Pages. nursing student I felt mixed emotions of excitement and anxiety when I was thinking about my upcoming clinical placement . Thesis! because even though I have been working as a patient care assistant at whale essays, Royal Darwin Hospital for a year and had already undergone clinical teaching block for one week . Thesis! The responsibility of being a nurse is big and much complex than my current job [1.2 Fulfils the duty of care] especially when handling assigned patients. Thus, need to have a good supervision from my clinical preceptor.

Health , Health care , Nurse 1455 Words | 4 Pages. timely and appropriately managed care, the nurse will need to possess and develop clinical reasoning and effective communication . skills. College! The significance of these skills will be outlined by discussing the importance of the thesis eur identification and management of early warning signs. Then an insight into the value of clinical reasoning for the nursing professional will be addressed. Arthur! Additionally, an thesis eur, overview of an effective communication technique between the nurse and other health care workers will be. Clinical trial , Communication , Health 2007 Words | 6 Pages. Nursing Students Education Experience A Critique of in india Male Nursing Students Education . Thesis Eur! Experience In this paper , the article titled “Men Student Nurses : The Nursing Education Experience” will be critiqued. This paper will discuss the writing college purpose of the article including the purpose of the thesis study, the whale essays research question, and eur, the central concepts. It will also discuss the research methods and design, the sample which includes the procedure for selecting the sample , inclusions. Nurse , Nursing , Original research 1662 Words | 5 Pages.

? In Summary a Brief History of the death of king arthur essay Advance Practice Nursing Jake Freeman University of eur St. Francis Abstract This paper . primarily explores the article from the Journal of Nursing called “Demographic Profiles of Certified Nurse -Midwives, Certified Registered Nurse Anesthetists, and coursework, Nurse Practitioners: Reflections on Implications for Uniform Education and Regulation”. It will also include insight from thesis, other resources in coursework doctorate support to the information provided by eur the article preceding. Advanced practice nurse , Healthcare occupations , Nurse 1254 Words | 6 Pages. Classroom”, I have learned about coursework doctorate programs three learning styles. There are three: Visual, Auditory, and Kinesthetic. In the article “The Effect of Learning Styles on thesis . Education and the Teaching Process” by coursework programs Ibrahium, he tells us that visual learning is where the student uses items such as charts, pictures, diagrams and animations to learn. Eur! He also explains further that auditory student’s use listening as means of learning and business cycle essay, kinesthetic learners use physical activities rather than watching and thesis, listening to a demonstration. Auditory learning , Education , Educational psychology 1097 Words | 3 Pages. Mr Jones‘ bed bath, and ensured that there were not any obstacles around the bed so that my colleague and I could work effectively. Treating people with . care, respecting their individuality and their dignity is a fundamental part of our conduct as nurses (The Code 2008), therefore we introduced ourselves to Mr Jones and explained what we were going to do even though he did not seem to understand.

As I closed the curtains around the the death of king essay bed, his behaviour began to change. He started using very harsh. Feeling , Miracles , Nurse 1932 Words | 6 Pages. PSY 101 2014 and eur, 2015 Reflection Paper Instructions and Samples 1. Relations-Gordon G. Fultz Reflection Papers These papers are in lieu of a Mid-Term and Final test and are designed . to, have you provide your thoughts and feelings about the class material covered. The success of thesis eur this class to individuals, and to the group as a whole, depends upon the extent to which you devote time reading and thinking about the material. The instructor expects each student to whale essays, read the text, listen carefully to the lectures and discussions and submit reflection papers with reactions to. Feeling , Friendship , Interpersonal relationship 2089 Words | 4 Pages.

and beliefs influence your response. Thesis Eur! In the third and final column, describe how much of your personal response to share in your reflection . paper . Business Cycle Essay! Ask yourself questions to guide your response. If you are struggling to gauge your own feelings or pinpoint your own response, try asking yourself questions about the experience or reading and how it relates to you. Sample questions might include:[3] Does the reading, lecture, or experience challenge you socially, culturally, emotionally, or theologically. 2005 albums , Academia , Conclusion 2344 Words | 7 Pages.

PSYCH 100 “ SELF REFLECTION ” As a . human who exist in this world I also feel what are my greatest strengths and where it comes from. My greatest strength is my willingness to learn and practice new things every day, so that I can keep my confidence and be better at the things that I face and do. Thesis Eur! I think of positive points when I try to whale essays, do something new, rather than thinking about the negatives, and focusing. Bible , Conceptions of thesis God , Friendship 1123 Words | 3 Pages. Self Reflection 1 Self -Assessment and, Reflection Paper Managerial Communication . (COM 515) February 6, 2001 Self Reflection 2 Self -Assessment and Reflection Paper In the last seven weeks, I had an opportunity to thesis eur, look back and whale essays, analyze the events that has shaped my life.

This was a unique experience where I was emotionally comfortable enough to eur, look back at life. I was able to coursework programs, objectively revisit many events that were often buried and too painful to face. I was free from the. Fear , Life , Negative feedback 1074 Words | 3 Pages. The Importance of Self-Care for Nurses.

The Importance of thesis Exercise and Other Self -Care Modalities for Nurses The profession of nursing requires a capacity . and joy for caring and healing others both mentally and physically. Nurses spend their careers caring for patients and their families often in writing reports college the worst and most frightening periods of eur their lives. Whale Essays! Nursing responsibilities can be lengthy, stressful and physically and emotionally demanding. The demands of the eur nursing profession coupled with the nursing shortage and longer. Allied health professions , Exercise , Health 2050 Words | 6 Pages. on Self -­?Observations and reports college, Self -­? Reflections Maximum 1000 words each term The purpose . of these three reports is to demonstrate how you have used the eur learning journal to coursework doctorate, develop your self -­?awareness from eur, term to term. Of King Essay! (See Section 11 about the Learning Journal) Each report is due at the first class meeting after each reading week. Guidelines: The Reports should include your reflections about. Awareness , Learning , Phenomenology 806 Words | 3 Pages. Reflection on Mentoring a Student.

critically reflect on the experience of mentoring and assessing a student in clinical practice. As the student . Thesis! was on a four week placement the assessment process will be discussed incorporating the the death of king essay qualities of the mentor and the effect it can have on the mentor/ student relationship. In turn evaluating the learning environment and teaching strategies used including learning styles, reflecting on how they were applied to thesis eur, help the student . Finally, the evaluation of overall performance as a mentor. . Assessment , Coaching , Education 2035 Words | 6 Pages. Violence Against Student Nurses and Newly Registered Nurses. Violence against student nurses and newly registered nurses Abstract Workplace violence toward . student nurses and newly registered nurses must be eliminated. College! Nursing students and new nurses are particularly vulnerable to acts of violence. Thesis Eur! This article discusses the status of the workplace violence of nurses , the effects of violence on life cycle students and thesis, new nurses , and some strategies can be used to reduce the workplace violence to nurses . Whale Essays! Keywords: student nurses , newly registered nurses , workplace violence . Abuse , Domestic violence , Health care 1063 Words | 3 Pages. an appreciation that each and every daily interaction augments my experience (Spouse 2003:200, Marris 1986 cited by Johns 2000:65). This enthusiasm, however, . has caused an eur, inhibitory effect on my self -directed researching, and created conflict in some placement areas.

Whilst developing my role as a nurse , my activities as a person at home and beyond have diminished, as I attempt to adjust to, the demands of both domains (Spouse 2003:109). I resent distracting influences, and frequently domestic pressures. Florence Nightingale , Knowledge , Learning 2040 Words | 6 Pages. Clinical Decision Making and end of life care. ? Clinical decision making and thesis eur, end of life care The purpose of this essay is to discuss important aspects of clinical . Thesis! intervention in Australia health care settings and end of life care. A vast number of the population in Australia is ageing and business cycle, numerous people suffer form chronic illnesses, they have more chances to die in the near future. As a consequence the thesis eur focus of the end of life care resources is concentrated in the elderly and chronic sufferers. Resources such as financial, human and.

Decision making , Health care , Health care provider 1655 Words | 6 Pages. ? Master's Prepared Nurse Interview Name: Vidia Saleem Grand Canyon University: Nurs 502 Feb. Reports College! 2nd, 2015 Top of Form Bottom of . Form Master's Prepared Nurse Interview Interview selection This paper is eur, about an interview of a master’s prepared nurse who is in the position of an the death of king arthur essay, Assistant Director of nursing. Thesis Eur! The theme of the interview is concerned in showing how a graduate of book reports college a Master in science in nursing would be able to prepare a nurse in assuming a bigger role in the nursing practice. Academic degree , Bachelor's degree , Healthcare occupations 1049 Words | 4 Pages. Integrating E-Technology in Nurse Education. Enhancing nurse education through technological advances is a growing trend of eur nursing programs today. As the essays crisis complexity and amount of thesis eur . information that nurses are required to comprehend, support has grown for integrated curriculum approaches that include instruction with the whale essays innovative use of technology. A new litany of eur teaching and learning methods has arrived to include internet clinical instruction, online discussion boards, and research opportunities for essays in india both students and thesis eur, faculty.

Programs. Education , Educational psychology , Nurse 1450 Words | 5 Pages. Introduction Reflection can be defined as “…reviewing experience from writing book college, practice that it may be described, analysed, evaluated and thesis, consequently . used to inform and change future practice.” (Bulman 2008:4). Study skills are important in whale essays university education as Cottrell (2013:7) states “… higher level study is different from their precious experience.”. Study skills enhance an efficient learning and lead you to success as a student . In this paper , I would like to show you the eur reflection on what I have. Cognition , Homework , Learning 1219 Words | 4 Pages. CLASS IX SCIENCE (THEORY) SUMMATIVE ASSESSMENT TERM II SAMPLE PAPER I Time: 3 hours General Instructions: i) ii) iii) iv) v) vi) . vii) viii) ix) The question paper comprises of two sections, A and, B, you are to attempt both the sections. All questions are compulsory. There is no overall choice.

However, internal choice has been provided in all the three questions of five marks category. Only one option in such questions is to be attempted. All questions of thesis eur section A and writing book, all questions of section B. Atom , Energy , Kinetic energy 1608 Words | 5 Pages. Clinical Governance Improving the Continuing Education of Nurses. Clinical Governance Improving the Continuing Education of Nurses – Myth or Reality? Nurse Management MSc . Practice Development Nursing Practice Route Faculty of Community Studies, Law and Education The Manchester Metropolitan University Jean Rogers Tutor: Mary Shaw Submission Date: 8th August 2005 Word Count: Nursing has changed radically over thesis eur, the last two decades and writing book reports, is continuing to thesis, do so.

Some would say for the better others for the worse (Rushford and Ireland 1997). The. Clinical trial , Health , Health care 802 Words | 3 Pages. Running Header: THE SELF PAPER The Self Paper (Nessa) The Self . Paper Society wonders why people are the way they are. Sometimes our surroundings and situations in life make us who we are. The following defines the whale essays self , self -concept, emotion, self -esteem, behavior, and thesis, self -presentation.

DEFINITION OF SELF According to, self is, “a combining form of the death arthur self and variously used with the meanings “of the self ” ( self -analysis ) and “by oneself or itself” ( self -appointed );. Conceptions of self , Identity , Individual 1395 Words | 5 Pages. Reflective: Patient and Student Nurse. ?In order to help me with my reflection I have chosen Gibbs (1988), as the model to help guide my reflective process. This model comprises of thesis a . process that helps the individual look at a situation and think about their thoughts and, feelings at the time of the incident. Reflective skills help us to think about what could have been done, so that if a similar situation occurs again the experience gained can be used to deal with the situation in eur a professional manner (Palmer et al 1997).

To enable. Communication , Florence Nightingale , James I of England 2204 Words | 4 Pages. Clinical Experience Reflection Journal Name: Date: Week 1. Essay! Choose one patient (new mother, newborn, or . pregnant/laboring woman) and identify the priority problem. Thesis Eur! What did you contribute toward resolving or easing the problem? N/A for this week 2. Provide an example of an incident during clinical that reinforced the theory you have learned in OB lectures. An incident that reinforced what I have learned in lecture was the postpartum hemorrhage simulation.

I learned how chaotic. Childbirth , Critical thinking , Goal 691 Words | 3 Pages. I will reflect on an incident that occurred during my shift to thesis eur, develop positive attitude towards nursing informatics. In this reflection , I am . going to use Gibbs (1988) Reflective cycle. This model is a recognized framework for my reflection . Gibbs (1988) consist of six stages to coursework doctorate, complete one cycle which is able to improve my nursing informatics and learning from the experience for better practice in eur future. The cycle starts with description of the situation, analysis of the whale essays feelings, evaluation. Health care , Health care provider , Illness 1794 Words | 5 Pages. Reflection Paper #1 Rhonda Carter Theology 104-D11 LOU April 15, 2013 I. Introduction . This paper will be written to discuss the two topics I have chosen to write about for my first reflection paper . The topics I chose are spiritual gifts and thesis eur, love. When you start to think about whale essays it, these two things go together.

If a person uses their spiritual gifts for the right reasons they will be doing it out of love for God. A lot of eur people these days do not use their. Bible , Christianity , God 1553 Words | 4 Pages. The Effects of Self -Esteem on Student Success Cristine Scott Central Michigan University CED 502 Student . Development in Higher Education Mr. John Laliberte In today’s’ society, success is often measured by academic and professional achievement. Coursework Doctorate Programs! Higher education provides more opportunity and freedom.

Statistics verify that generally, the thesis eur more highly educated have higher earnings and there is a significant difference between wages earned by employees with College degrees and those without. Academic degree , College , Education 1498 Words | 5 Pages. Reflection: Surgery and book reports college, Reflective Practice. discusses a reflective practice with regards to a clinical placement I undertook. In the following critical incident that I encountered I will . utilize the Gibbs Reflective Model. Gibbs reflective model is thesis eur, fairly straightforward and encourage a clear description of the situation. Analysis of feelings, evaluation of the experience, analysis to essays on water crisis, make sense of the thesis experience, conclusion and action plan where other options are considered and reflection upon experience to examine what you would do if the. Health care , Patient , Physician 1119 Words | 3 Pages. Student -Preceptor Relationships Meaningful Event The purpose of my paper is to reflect upon the . student -preceptor relationship and how it shapes the views and interests of the novice nurse within the nursing profession.

Thinking back, clinical opportunities that were meaningful to me have been when I built a positive and professional relationship with my preceptors. These well developed relationships have directly correlated with how well my overall clinical experience has gone and business essay, they’ve shaped. Florence Nightingale , Nurse , Nurse uniform 1852 Words | 5 Pages. The Metamorphosis of the eur Practical Nurse. The Metamorphosis of the Practical Nurse October 23, 2006 The Metamorphosis of the Practical Nurse It takes a great deal . of courage to let go of what is known, familiar, and comfortable. Change is coursework doctorate, a driving force in everyone. Like a butterfly, individuals in the nursing field may go through similar stages of metamorphosis, which is a process of thesis growth, change, and development. (Wikipedia Foundation, 2006) Nurses have a vast amount of opportunity for growth and change in on water crisis in india the healthcare. Florence Nightingale , Healthcare occupations , Leadership 1810 Words | 5 Pages.

THE TRANSITION STUDENT TO GRADUATE NURSE. FROM STUDENT TO A GRADUATE NURSE The new graduate nurses (NGN) are faced with various issues and eur, challenges . especially in their first year of essays on water in india nursing practice. The period of transition from a student to a graduate nurse is a demanding period that is filled with new experiences and there are several concerns and factors that can affect the transition process. The research into thesis the issues has recommended some strategies that can be utilised to ease the business transition process from being a student to a. Bachelor of thesis eur Science in Nursing , Healthcare occupations , Licensed practical nurse 2349 Words | 7 Pages. Differences in Competencies Between Adn and Bsn Nurses. between ADN and BSN nurses GRAND CANYON UNIVERSITY Professional Dynamics NRS 430v January 07, 2012 Differences in competencies . Reports College! between ADN and BSN nurses Registered nurses comprise the largest sector in the health care field. 2008 National Sample Survey of Registered Nurses conducted by the Health Resources And Services Administration (HRSA) says that nursing is one of the nation's biggest health Care Profession and it comprises more than 3 million registered nurses (RNs) in the United.

Academic degree , Associate's degree , Bachelor of Science 1309 Words | 5 Pages. Critique of the Study by Sohn and thesis, Colleagues, “Simulation-based smoking cessation intervention education for undergraduate nursing students” Critique of the Study by Sohn and Colleagues, “Simulation-based smoking cessation intervention education for undergraduate nursing students ” . In this paper , Sohn and colleagues (2011) study Simulation-based smoking cessation intervention education for whale essays undergraduate nursing students will be critiqued with the assistance of Loiselle and Profetto-McGrath’s (2011) book Canadian Essentials of Nursing Research. Title The title of the article is clear and concise. It is only ten words long and clearly. Albert Bandura , Nursing , Nursing school 2305 Words | 7 Pages. This reflection will discuss personal diffidence and how it has influenced my academic studies, including my practice within placement . settings.

Gibbs reflective cycle (Gibbs, 1998) has been utilised as it illustrates a clear structure for the process of thesis eur reflection . To conclude this reflection will draw together the themes which have emerged and highlight a clear action for writing book reports future learning that will be used in order to thesis eur, enhance my future practice.! Within my previous occupation as a health care assistant. Assertiveness , Feeling , Future 1710 Words | 5 Pages.

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to thesis eur which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group.

Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Arthur Essay! Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand.

Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . . . . Thesis Eur! . . . . . . . . . Coursework! . . . . . . . . Thesis Eur! . . . Whale Essays! . . . . . . 5 Prerequisites . . . Thesis Eur! . . . . . . . . . . . . . . Coursework Programs! . . . . . . . . . . . . . . . . . . . . 5 Acknowledgements . . . . . . Eur! . .! . . . . . . . . . . . . . . Thesis! . . . . . . . . . . . 5 Introduction . . Whale Essays! . . . . . . . . . . Eur! . .! . . . . . Thesis Eur! . . . . . . . . . . . . Coursework Doctorate Programs! . . . . . . 1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis Eur! . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . Whale Essays! . Eur! . . . . . . . . . . Writing Book Reports! . . . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . . . . . . . . . Eur! . . .! . . . Eur! . . Whale Essays! . Thesis Eur! . . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . Business Essay! . . . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Local rings . . . . . Thesis! . . . . . . . . . . . . In India! . . . . . . . . . . . . . . . . . . . . 10 Rings of fractions . . Eur! . . . . . . . . . . . . . . . . . . . . . . The Death Of King Arthur Essay! . . . . . . Thesis! . . . . 11 The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . .! . Thesis! . . . . . 12 Review of tensor products . . . . . . . . . . . . . . . . Whale Essays! . . . . . . . . Thesis! . Essays On Water Crisis In India! . . . . . 14 Exercise . Thesis! . . . . . . . . . . . Coursework Doctorate Programs! . . . . . . . . . . . . . . . . . . . . . Thesis! . . . . . . Whale Essays! 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . . . . . Eur! . . . . . 19 Dedekind’s proof that the business essay, integral elements form a ring . . . . . . . . . . . . . . 20 Integral elements . . . . . . . . . . . . . . . . . . . . . Thesis Eur! . . . . . . . . . . Coursework Doctorate! . . . 22 Review of bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . Thesis Eur! . . . . 25 Review of norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Review of bilinear forms . . . . . . Whale Essays! . . . . . . . . . . . . . Eur! . . . . . . . . . . . 26 Discriminants . . . . . . . . Reports! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rings of integers are finitely generated . . . . Eur! . . . . . . . . . . . . . Of King Arthur Essay! . Thesis Eur! . . . . . 29 Finding the ring of integers . . . . . . . . . . . . . . . . . .! . . . . . . . . . . . 31 Algorithms for thesis, finding the ring of integers . . . . . . . . . . . . The Death Of King! . . . . Eur! . . . . .! 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . . . . . . . . . . Thesis Eur! . . . . . . . . . . . . . . . . 40 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . Life Cycle! . . . . . . . . . . . . . . . Thesis Eur! . . Coursework Doctorate Programs! . . . . . . . 43 The ideal class group . . . . . Thesis! . . . . . . . . . . . . . . . . . . . . . . . . . . . The Death! 46 Discrete valuations . . . . . Thesis! . . . . . . On Water Crisis! . . . . . . . . . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . . . . . . . . . . . . . . . . Thesis Eur! . . 51 Modules over Dedekind domains (sketch). . . . In India! . . . . . . . Eur! . . . . . . . . . . . Whale Essays! 52. Eur! Factorization in extensions . . . . . . . . . . . . . . Essays On Water In India! . . . . . . . . . . . . . . . 52 The primes that ramify . . . . . . Thesis! . . . . . . . . . . . Life Cycle! . . . . . . . . . . . . . . 54 Finding factorizations . . . . . . . . . . . . . . . . . . . Thesis Eur! . . . . . . . . . Whale Essays! . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . Thesis Eur! . . . . . . . . . . . . 57 Eisenstein extensions . . . . . . . . . . . . . . . . Coursework Doctorate Programs! . . Eur! . . . . . . . . . . . . . . 60 Exercises . . . Essays Crisis! . . . . . . . . . . . Thesis! . . . . . . . Essays On Water Crisis! . . . . Thesis! . Life Cycle Essay! . . . Eur! . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . .! . . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis! 63 Statement of the main theorem and its consequences . . . . . . . Book Reports! . . . . Thesis! . . . . . 65 Lattices . . .! . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis Eur! . . . . . . . . . . 68 Some calculus . . . . . . . . . . . . . . . . On Water Crisis In India! . . Thesis! . . . . . . . . . . . . . . . . . . 73 Finiteness of the class number . . . . . . . . . . Essays Crisis! . . . . . . . . . . . Eur! . . . . . . 75 Binary quadratic forms . . . . . . . . . . . . . . . . . . Coursework Programs! . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . Thesis! . . . Whale Essays! . . . . . . . . . . . . . Thesis! . . . . . 78. Essay! 5 The Unit Theorem 80 Statement of the thesis eur, theorem . . . . . . . . . . . . . . Writing! . . Thesis! . . Whale Essays! . . . . . . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . Eur! . . . . . Writing Reports! . . . . . . . . . . . . 82 Computation of the rank . . . . . . . . Thesis! . . . Book College! . . . . . . . . . . . . . . . . . . . 83 S -units . Eur! . . . . . . . . . Business! . Thesis! . . . . . . . . The Death! . . . . . . . . . . . Thesis Eur! . . . . . . . . . . 85 Example: CM fields . Book! . . . . . . . . . . . . . . . . . . . . . . . Thesis! . . . . . . . . . 86 Example: real quadratic fields . . . . . . . . Coursework Doctorate Programs! . Thesis Eur! . . . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . . . 87 Finding .K/ . . . . . . . . . . . . Life Cycle Essay! . . . . . . . . . . . . . . . . Eur! . . . . . Of King Arthur! . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . . . . Thesis! 89 Regulators . . Book College! . Thesis Eur! . . . . . . . . . . . . . . . . . . . Coursework Programs! . . . . . . . . . . . . . . . . 89 Exercises . . . . . . . Thesis! . . . . . . . . . . Whale Essays! . . Eur! . . . . . . . . . . . . Essays Crisis In India! . . . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eur! . . . 91 Class numbers of cyclotomic fields . . Coursework Doctorate Programs! . . . . . . . Thesis Eur! . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . On Water In India! . . . . . . . . . . . Eur! . . . . . . . . . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . . Life Essay! . . . . . . . 98 Exercises . . . . . . Eur! . . . . . . . . . . . . . Essays On Water! . . . . . . . . . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . Eur! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Business Life Essay! . . Thesis Eur! . Essays In India! 101 Nonarchimedean valuations . . . . . . . . Thesis Eur! . . . . . . . . .! . . . . . Eur! . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . Doctorate Programs! . Thesis Eur! . . 105 Complete list of valuations for the rational numbers . . . . . . . . . . . . . . . . 105 The primes of a number field . . Essays On Water Crisis In India! . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . Thesis Eur! . . . . . Whale Essays! . . . Thesis Eur! . .! . . . . . Eur! . . . 109 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . Coursework Doctorate! . . . . . . . . . Eur! . . . . . . 111 Newton’s lemma . . . . Doctorate! . . . . . . . . . Thesis! . . Writing Book College! . . . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . Thesis! . . . . . . . 118.

Newton’s polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Locally compact fields . . . . . . . . . . . . . . . . . . . . . Writing College! . . . . . . . . . . 122 Unramified extensions of a local field . . . . Thesis Eur! . . . . . . . . . . . . Writing Book College! . . . Thesis! .! . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Ramification groups . . . . . . . . . . Eur! . . . Life Cycle! . . . . . . . . . . . . . . . . . . . . 126 Krasner’s lemma and applications . . . . . . . . . . . . Eur! . . . . . The Death Of King Arthur! . . . . . . . . 127 Exercises . . . . . . . Eur! . . . . . . . . . . . . . . . . . . . . . . . . . . . Doctorate Programs! . . . . 129. Thesis Eur! 8 Global Fields 131 Extending valuations . . . .! . Thesis! . . . . . . . . . . . . . . . Whale Essays! . . Thesis! . . . . . . . . . . 131 The product formula . . . . . . . . . . Coursework! . . . . . . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . Thesis Eur! . .! . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . Thesis! . . . . .! . . . Eur! . . . . . . . . . . . . . . . . . . . 137 Examples . . . . . . . .! . Thesis! . . . . . . .! . . . Thesis! . . . . . . . . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . Doctorate Programs! . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . Thesis Eur! . . . Book Reports! . . . . . . . Eur! . . . . . . . . 141 Applications of the Chebotarev density theorem . . . Whale Essays! . . . . . . . . . . . . . . . 146 Exercises . . . Thesis! . Book Reports! . . . . . . . . . . . . . . . . . . Eur! . . . . . . . . . . . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number.

For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is reports college a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by thesis eur, jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by the death, definition; X Y X is isomorphic to Y ; X ' Y X and eur Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q. The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful.! In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at FT Fields and Galois Theory, v4.22, 2011.

GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. Thesis! He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?. The English mathemati- cians found an algorithm for solving the arthur, problem, but neglected to prove that the algorithm always works.

EULER (1707–1783). He introduced analysis into thesis eur, the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and doctorate n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and eur used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of, a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810–1893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes. HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions.

EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for eur, generating abelian extensions of number fields (the Kronecker liebster Jugendtraum). RIEMANN (1826–1866). Studied the Riemann zeta function, and the death of king essay made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the thesis, ring of integers in a number field, by the death of king arthur, proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into thesis, finitely many classes.

Defined the zeta function of a number field. WEBER (1842–1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941). Life Cycle! He gave the first definition of the field of thesis eur, p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943).

He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). On Water In India! He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and thesis eur Hilbert. NOETHER (1882–1935). Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over business life number fields.

HECKE (1887–1947). Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions. ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979). He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to thesis the classification of central simple algebras over whale essays number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of eur, central simple algebras over number fields. WEIL (1906–1998).

Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925– ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for of king arthur essay, a given number, does not also belong to thesis the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is the death essay a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields.

Three problems present themselves. First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. Finally, since factorization is only considered up to units, in thesis order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the of king, algebraic integers form a subring OK of thesis eur, K. The criterion as stated is difficult to on water in india apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?.

Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P. as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of thesis eur, irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain? No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have.

6D 2 3D .1C p ?5/.1?.! To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and. 1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. Thesis! In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is coursework unique if all irreducible elements are prime. What can we recover?

Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of thesis eur, “prime numbers” so that, for example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”. How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Book Reports! Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”.

Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of thesis eur, you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. Of King Essay! .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the thesis, four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime. For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so.

Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the essays crisis in india, form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index. The index is called the thesis eur, class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units.

For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at thesis eur the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that. Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction.

Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?. Essays Crisis In India! Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. Thesis! By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and essays Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY. Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is thesis a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory.

Lemmermeyer 2009, which explains the origins of “ideal numbers”, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is whale essays as described. 0-2 Complete the eur, verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals.! CHAPTER 1 Preliminaries from Commutative. Many results that were first proved for rings of integers in number fields are true for more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the identity element to the identity element. A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the smallest subring of B containing A and thesis eur the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . Writing Reports College! ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the thesis eur, ai . It consists of finite sums. Writing Book Reports! P ciai , ci 2 A, and it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. Thesis Eur! aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . Thesis Eur! ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . Coursework Doctorate! ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA! A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the thesis, ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. The Death Of King Arthur Essay! An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to eur be prime if ./ is a prime ideal; equivalently, if jab) ja or jb.

An ideal m in A is maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field.! Every proper ideal a of A is contained in a maximal ideal — if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorn’s lemma. In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in eur products of rings.

PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in coursework doctorate A and thesis eur B respectively, then ab is an ideal in AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the form. pB (p a prime ideal of A), Ap (p a prime ideal of B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an whale essays, integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is thesis eur a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j: A ring A is Noetherian if every ideal in A is finitely generated.

PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. Book Reports! From (b) we know that this process will lead to a maximal element after only eur finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the arthur essay, zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/. If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the thesis, rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements. PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and whale essays is such that .a/ is maximal among the ideals generated by thesis, such elements (here we use that A is essays in india Noetherian). Thesis! Since a can not be written as a product of, irreducible elements, it is not itself irreducible, and thesis so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for life cycle, c. Thus a is thesis a product of irreducible elements, and we have a contradiction. 2. Coursework Programs! REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of thesis, Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an irreducible element of life cycle, O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated. PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF.

Similar to the proof of Proposition 1.3. 2. Thesis Eur! PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant. Now the essays on water crisis, claim shows that the chain itself becomes constant. 2. Thesis! PROPOSITION 1.8 Let A be a Noetherian ring. Then every finitely generated A-module is Noetherian.

PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by cycle, induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for thesis, then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions. Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. For example, Q is the field of fractions of Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define.

S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg: 2This notation conflicts with a later notation in which Zp denotes the of king essay, ring of p-adic integers. Thesis! 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an coursework doctorate, integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of S?1A aec D a if a is a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in S?1A. For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Thesis Eur! Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of prime ideals in on water crisis S?1A; the inverse map is p 7! pA. PROOF.

It is easy to see that. Thesis Eur! p a prime ideal disjoint from S) pe is of king a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse. 2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in general, for t a nonzero element of an integral domain, fprime ideals of thesis, Atg $ fprime ideals of A not containing tg. fprime ideals of A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to thesis be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals.

Q .mi /. If the mi are relatively prime in. Whale Essays! pairs, then mD Q mi , and so we have. Q .mi /. Note that in thesis general, a1 a2 an a1a2 . an; but the whale essays, two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence. PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. Thesis! The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q Cbi /D 1, and lies in a1C. Q i2 ai , and so.

We can now apply the theorem in the case nD 2 to obtain an business life cycle essay, element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. Thesis! For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an essay, A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ! P A-linear. The tensor product exists, and thesis eur is unique (up to a unique isomorphism making the obvious diagram commute).

We denote it by M ?AN , and we write .m;n/ 7! m?n for reports college, f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is thesis of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus.! BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is thesis a canonical isomorphism. Review of tensor products. It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and business life cycle n respectively, then V ?kW is a vector space over k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that.

REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the thesis, mr ns matrix (linear map kns! kmr ) with. [email protected] a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is whale essays generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it. 2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module.

Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by thesis, about 70 years. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism.

M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS. Writing Book Reports! If A! B is an A-algebra and M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by thesis eur, extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is business life essay a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. Thesis! TENSOR PRODUCTS OF ALGEBRAS. If f WA! B and gWA!

C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. The Death Of King! .b? c/.b0? c0/D bb0? cc0. and thesis the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of, a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the thesis eur, minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Coursework Programs! Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X? .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of thesis eur, ?, If ? is a primitive element for K=k, then the image ?i of coursework programs, ? in eur ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in coursework doctorate (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. Thesis Eur! It was not until the 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on your website.

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40 Useful Words and eur Phrases for Top-Notch Essays. The secret to writing reports, a successful essay doesn’t just lie in the clever things you talk about and thesis the way you structure your points. To be truly brilliant, an life, essay needs to thesis eur, utilise the right language. You could make a great point, but if it’s not intelligently articulated, you almost needn’t have bothered. Developing the language skills to build an argument and to write persuasively is writing reports college, crucial if you’re to write outstanding essays every time. In this article, we’re going to eur, equip you with the words and phrases you need to write a top-notch essay, along with examples of how to writing book reports college, utilise them. Thesis Eur? It’s by no means an exhaustive list, and there will often be other ways of coursework, using the words and phrases we describe that we won’t have room to include, but there should be more than enough below to help you make an instant improvement to your essay-writing skills – whether you’re a native English speaker or taking your first steps into writing essays in English. Let’s start by looking at thesis, language for general explanations of coursework, complex points.

Usage : “In order to” can be used to introduce an thesis eur, explanation for the purpose of an argument. Example : “In order to understand X, we need first to writing, understand Y.” Usage : Use “in other words” when you want to express something in a different way (more simply), to make it easier to thesis eur, understand, or to emphasise or expand on a point. Example : “Frogs are amphibians. In other words, they live on thesis eur, the land and in the water.” Usage : This phrase is another way of saying “in other words”, and can be used in particularly complex points, when you feel that an alternative way of wording a problem may help the reader achieve a better understanding of its significance. Example : “Plants rely on thesis, photosynthesis. To put it another way, they will die without the sun.”

Usage : “That is” and “that is to, say” can be used to add further detail to your explanation, or to be more precise. Example : “Whales are mammals. Eur? That is to say, they must breathe air.” Usage : Use “to that end” or “to this end” in a similar way to “in order to” or “so”. Example : “Zoologists have long sought to understand how animals communicate with each other. To that end, a new study has been launched that looks at elephant sounds and their possible meanings.” Adding additional information to eur, support a point. Students often make the mistake of using synonyms of “and” each time they want to add further information in on water crisis in india support of a point they’re making, or to build an argument. Here are some cleverer ways of doing this.

Usage : Employ “moreover” at the start of a sentence to eur, add extra information in support of a point you’re making. Example : “Moreover, the results of a recent piece of research provide compelling evidence in support of…” Usage :This is also generally used at the start of a sentence, to, add extra information. Example : “Furthermore, there is evidence to suggest that…” Usage : This is used in the same way as “moreover” and “furthermore”. Example : “What’s more, this isn’t the only evidence that supports this hypothesis.” Usage : Use “likewise” when you want to talk about something that agrees with what you’ve just mentioned. Example : “Scholar A believes X. Thesis Eur? Likewise, Scholar B argues compellingly in favour of this point of view.”

Usage : Use “similarly” in the same way as “likewise”. Example : “Audiences at coursework doctorate, the time reacted with shock to eur, Beethoven’s new work, because it was very different to what they were used to. Similarly, we have a tendency to coursework programs, react with surprise to the unfamiliar.” Usage : Use the phrase “another key point to remember” or “another key fact to remember” to introduce additional facts without using the eur word “also”. Example : “As a Romantic, Blake was a proponent of a closer relationship between humans and nature. Another key point to remember is that Blake was writing during the whale essays Industrial Revolution, which had a major impact on the world around him.”

Usage : Use “as well as” instead of thesis eur, “also” or “and”. Example : “Scholar A argued that this was due to X, as well as Y.” Usage : This wording is used to business life, add an extra piece of information, often something that’s in some way more surprising or unexpected than the first piece of thesis, information. Example : “Not only did Edmund Hillary have the honour of being the first to reach the summit of Everest, but he was also appointed Knight Commander of the Order of the British Empire.” Usage : Used when considering two or more arguments at a time. Example : “Coupled with the literary evidence, the statistics paint a compelling view of…” Usage : This can be used to structure an argument, presenting facts clearly one after the other. Example : “There are many points in whale essays support of this view.

Firstly, X. Secondly, Y. And thirdly, Z. 16. Not to mention/to say nothing of. Usage : “Not to thesis, mention” and “to say nothing of” can be used to writing book college, add extra information with a bit of emphasis. Example : “The war caused unprecedented suffering to millions of people, not to eur, mention its impact on the country’s economy.” Words and phrases for demonstrating contrast. When you’re developing an argument, you will often need to present contrasting or opposing opinions or evidence – “it could show this, but it could also show this”, or “X says this, but Y disagrees”.

This section covers words you can use instead of the “but” in these examples, to make your writing sound more intelligent and interesting. Usage : Use “however” to coursework doctorate, introduce a point that disagrees with what you’ve just said. Example : “Scholar A thinks this. However, Scholar B reached a different conclusion.” Usage : Usage of this phrase includes introducing a contrasting interpretation of the same piece of thesis, evidence, a different piece of evidence that suggests something else, or an programs, opposing opinion. Example: “The historical evidence appears to suggest a clear-cut situation.

On the other hand, the archaeological evidence presents a somewhat less straightforward picture of what happened that day.” Usage : Used in thesis eur a similar manner to business cycle essay, “on the other hand” or “but”. Example : “The historians are unanimous in telling us X, an eur, agreement that suggests that this version of events must be an business, accurate account. Thesis Eur? Having said that, the essays archaeology tells a different story.” Usage : Use “by contrast” or “in comparison” when you’re comparing and eur contrasting pieces of business cycle essay, evidence. Example : “Scholar A’s opinion, then, is eur, based on essays on water crisis, insufficient evidence. By contrast, Scholar B’s opinion seems more plausible.” Usage : Use this to thesis eur, cast doubt on an assertion. Example : “Writer A asserts that this was the reason for what happened. Then again, it’s possible that he was being paid to say this.”

Usage : This is used in the same way as “then again”. Example : “The evidence ostensibly appears to point to this conclusion. That said, much of the evidence is the death essay, unreliable at best.” Usage : Use this when you want to introduce a contrasting idea. Example : “Much of scholarship has focused on this evidence. Yet not everyone agrees that this is the most important aspect of the situation.” Adding a proviso or acknowledging reservations. Sometimes, you may need to acknowledge a shortfalling in a piece of evidence, or add a proviso.

Here are some ways of eur, doing so. Usage : Use “despite this” or “in spite of this” when you want to book, outline a point that stands regardless of a shortfalling in the evidence. Example : “The sample size was small, but the thesis eur results were important despite this.” Usage : Use this when you want your reader to consider a point in the knowledge of something else. Example : “We’ve seen that the whale essays methods used in the 19th century study did not always live up to the rigorous standards expected in scientific research today, which makes it difficult to thesis eur, draw definite conclusions. With this in mind, let’s look at a more recent study to see how the results compare.” Usage : This means “on condition that”.

You can also say “providing that” or just “providing” to, mean the eur same thing. Example : “We may use this as evidence to support our argument, provided that we bear in mind the limitations of the methods used to obtain it.” Usage : These phrases are used when something has shed light on life essay, something else. Example : “In light of the evidence from the 2013 study, we have a better understanding of…” Usage : This is similar to “despite this”. Example : “The study had its limitations, but it was nonetheless groundbreaking for its day.” Usage : This is the same as “nonetheless”.

Example : “The study was flawed, but it was important nevertheless.” Usage : This is eur, another way of saying “nonetheless”. Example : “Notwithstanding the limitations of the methodology used, it was an, important study in the development of thesis eur, how we view the workings of the human mind.” Good essays always back up points with examples, but it’s going to, get boring if you use the thesis expression “for example” every time. Here are a couple of other ways of saying the coursework programs same thing. Example : “Some birds migrate to avoid harsher winter climates. Thesis Eur? Swallows, for instance, leave the UK in writing early winter and thesis fly south…” Example : “To give an illustration of what I mean, let’s look at the case of…” When you want to the death arthur essay, demonstrate that a point is thesis, particularly important, there are several ways of cycle, highlighting it as such. Usage : Used to introduce a point that is loaded with meaning that might not be immediately apparent. Example : “Significantly, Tacitus omits to tell us the kind of gossip prevalent in Suetonius’ accounts of the same period.”

Usage : This can be used to mean “significantly” (as above), and it can also be used interchangeably with “in particular” (the example below demonstrates the first of these ways of using it). Example : “Actual figures are notably absent from Scholar A’s analysis.” Usage : Use “importantly” interchangeably with “significantly”. Example : “Importantly, Scholar A was being employed by X when he wrote this work, and thesis eur was presumably therefore under pressure to whale essays, portray the situation more favourably than he perhaps might otherwise have done.” You’ve almost made it to the end of the essay, but your work isn’t over yet. You need to end by wrapping up everything you’ve talked about, showing that you’ve considered the arguments on thesis eur, both sides and reached the book most likely conclusion. Here are some words and phrases to help you.

Usage : Typically used to thesis, introduce the concluding paragraph or sentence of an essay, summarising what you’ve discussed in a broad overview. Example : “In conclusion, the on water in india evidence points almost exclusively to Argument A.” Usage : Used to signify what you believe to be the most significant point, and the main takeaway from the essay. Example : “Above all, it seems pertinent to remember that…” Usage : This is thesis eur, a useful word to use when summarising which argument you find most convincing. Example : “Scholar A’s point – that Constanze Mozart was motivated by financial gain – seems to me to be the coursework programs most persuasive argument for her actions following Mozart’s death.”

Usage : Use in the same way as “persuasive” above. Example : “The most compelling argument is eur, presented by whale essays, Scholar A.” Usage : This means “taking everything into account”. Example : “All things considered, it seems reasonable to assume that…” How many of eur, these words and phrases will you get into your next essay? And are any of your favourite essay terms missing from our list?

Let us know in coursework doctorate the comments below! 221 Responses to “40 Useful Words and Phrases for thesis eur, Top-Notch Essays” January 09, 2015 at, 8:47 am, Jimmy Tan said: January 23, 2016 at 1:13 am, AN INDIAN said: It is eur, very useful for junior as well as to senior.It is awesome……………… thanks for, this. November 29, 2016 at 9:46 am, Mofasa said: March 10, 2017 at 3:30 pm, Vaibhavi said:

May 04, 2017 at 3:54 pm, felix said: September 17, 2017 at 8:02 am, arjun said: April 29, 2016 at thesis, 1:13 pm, ron said: September 01, 2016 at 1:18 am, RedKeyMon said: Ty bruv! It’s really useful! November 26, 2016 at 12:00 pm, Ranch Dressing said: This the real RedKeyMon? May 26, 2017 at 2:28 pm, inosh said:

September 07, 2016 at 12:04 am, winter the savage said: yeh these words are lit. October 15, 2016 at 6:19 pm, Freya said: Thanks!! I have to right an whale essays, essay on the battle of hastings and I really need to thesis eur, widen my vocab! It really helped December 12, 2016 at arthur essay, 1:13 pm, rose said: great thanks,now i know how to use big words in thesis an argumentative essay. April 18, 2017 at 4:55 pm, nikie said: very very useful! May 26, 2017 at 9:13 am, Anthony said:

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You are correct. The Death Essay? I have taught writing for thesis eur, 40-plus years, and I find many of these suggestions wordy and unnecessary. For instance, “In order to” can simply to of king arthur essay, be “To.” June 02, 2015 at 1:48 am, phertauwete said: Thanks so much, I got it. June 05, 2015 at 5:17 am, william said: Thank you very much, I’m having my O-level English in about 2 hours now and you really helped me. I’m sure i can score an thesis, A now #128512; June 07, 2015 at coursework doctorate programs, 7:21 am, Lysha said: Great!! Thank you so much for thesis eur, including the of king arthur essay tips above.

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Thanks for the list. I think it would also be helpful to include a list of words to eur, help express the thoughts of the writers being referenced in the essay, for example, “Jacob states that teachers need to consider the learning abilities of essays in india, all students” or “Williams asserts that reading English texts more often can increase the thesis eur writing skills of ESL learners”. Whale Essays? It sometimes becomes boring to keep saying, “so-and-so states that…” October 12, 2015 at 9:07 am, faiz said: This was useful for thesis eur, me. October 14, 2015 at 3:09 am, Lilow W. said: This article was wonderful. Furthermore, I don’t think my world history grade, as well as my grades in language arts and science, would have been half of what they are now without it. All things considered, this was an amazing article and cycle essay I recommend it to thesis, anyone looking to give their essay an elegant twist. October 15, 2015 at 3:41 pm, Ayushi said:

Thanks!! Really a very useful list! October 18, 2015 at 3:29 pm, Missy said: Really helpful ……………. Business Cycle? Thank you guys. October 19, 2015 at 10:49 am, steven O'Donnell said: I would have also included “in fact” October 19, 2015 at 9:42 pm, Tom Howell said: *Gasp* As someone who has studied and takes a modicum of interest in the English language, and having spent a lifetime writing scientific essays, all I can say is this list of cliches will not aid you in your academic pursuits. Eur? This is a categorical list of English don’ts that will undoubtedly be highlight with a sigh, and a comment something along the arthur essay lines of “Please, no more” – paraphrased of eur, course. Some of these are inescapable and are useful tools, but please use in writing moderation. If you truly wish to improve your writing, consider the use of language in thesis literature you may study as part of your academic endeavours – be it english or technical based.

What makes a writing style worthy of note, and what makes it tedious and monotonous? The use of structure, engaging writing styles, and even metaphors can be true keys to essay success. December 07, 2015 at business cycle essay, 8:24 am, Jeff B. Thesis Eur? said: Tom, I must agree. In my English class such cliches as listed here would be highlighted and returned to the student with an admonition to “please use your own words,” or something to that effect. As I like to joke, “Cliches should be avoided like the whale essays plague.” October 20, 2015 at eur, 8:22 pm, yoursif said: October 22, 2015 at 5:00 pm, nur amira said: thanks it was indeed helpful.

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I don’t really feel comfortable using firstly, secondly, thirdly to introduce new points… does anyone else feel the same? December 07, 2015 at 7:58 am, Jeff B. said: While some of the suggestions in this article are valuable, I am going to have to agree with Tom Howell about the use of cliches. Eur? Many of the phrases suggested (e.g. “To put it another way” and “Another key thing to remember”) are cliches, and whale essays should be avoided in original writing. Another note: this article recommends the use of “In conclusion” to introduce the thesis eur conclusion. My suggestion is to avoid it, and I am not the only one who feels this way. To quote the Writing Center at Harvard University: summary,’ and ‘to sum up.’ These phrases can be. useful–even welcome–in oral presentations. But.

readers can see, by the tell-tale compression of the. pages, when an essay is about to end. You’ll irritate. your audience if you belabor the obvious.” If your readers cannot figure out that from reading your text that you are presenting your conclusion, then your conclusion has more problems than simply the thesis introduction to whale essays, it. December 07, 2015 at 8:03 am, Jeff B. said: Note: In my comment above, I should have specified “for college-level writing… and above.” Further, I will say that sometimes, indeed, “In conclusion” can be a useful phrase, but in eur many if not most cases it should be avoided. December 07, 2015 at 10:43 am, Jeff B. said: * Assuming, of course, that my “comment above” is actually posted after it undergoes “moderation.” Otherwise my note makes no sense… December 07, 2015 at coursework doctorate, 8:25 pm, pavla said: put (or set) the record straight.

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IGCSE and GCSE Geography Coursework (Paper 4) Paper 4 is the alternative to coursework paper that is worth 27.5% of your final IGCSE grade. The coursework paper tests your understanding of how coursework is carried out. Thesis? The paper is 1 hour and whale essays, 30 minutes long and divided into thesis eur two 30 mark questions. Crisis In India? You answer all the questions on the examination paper, most questions are short in length, between 1 and 4 marks. Thesis? The paper requires a combination of knowledge and book reports college, skills.

The questions usually focus on coursework to do with; coasts, rivers, settlements or weather. As a IGCSE geographer it is possible to carry out coursework on many aspects of the course. Thesis Eur? Possible topics and coursework doctorate, areas of study may include: Changes in river velocity from source to mouth Changes in thesis, load (shape and size) from source to whale essays, mouth Changes in eur, channel depth, width, cross-section and wetted perimeter from source to mouth Changes in discharge from source to mouth Changes across a meanders cross section Changes in river gradient from of king essay, source to mouth Changes in valley size and gradient from source to mouth Changes in land use a long a river Changes in pollution along a rivers' course (you need proper equipment for this) Changes in vegetation (cover and eur, variety) moving inland Changes in beach profile and sand dune profile Speed of longshore drift Changes in land use Changes in defences (compare to land use) Changes in beach or dune material (size or shape) Changes in land use Changes in traffic (maybe CBD to rural-urban fringe) Changes in on water, pedestrians Number and type of tourists Changes in the quality of the environment or pollution levels (may look at the impact of industry) Changes in globalisation Changes in thesis eur, cost of products Comparison to of king essay, Burgess or Hoyt Model Changes in wealth or population density (will have to use some secondary data) Sphere of influences of settlements or services. Changes in temperature throughout the day or between seasons Changes in rainfall Changes in eur, humidity Changes in hours of sunshine or cloud cover Changes in wind speed and wind direction. Before you choose your coursework you have to decide if you live in a suitable study location. For example it is whale essays, no use trying to thesis, do a piece of essays, coursework on rivers, if you live in the middle of a desert. Before starting your coursework, you should also think about eur, how you can carry out the writing book coursework safely and definitely carry out a risk assessment. You can make your coursework safer by doing the following:

Protection from the weather (waterproof jacket, umbrella, hat, suncream) Sensible dress (remember you will be representing your school, but you should also wear clothes that don't draw attention to thesis, yourself Always carry out coursework in groups Always tell an adult or teacher where you area carrying out coursework Always carry a mobile phone with you Never do coursework near a river or the sea without an adult or teacher and without them checking that it is safe Carry out coursework in day light and wear reflective clothes Check that your study area is writing reports college, safe. For example it wouldn't be safe walking around downtown San Salvador Don't display valuables making you more vulnerable to crime e.g. if you have a camera or a phone keep it out of sight. Specification: Formulating aims and hypotheses: Candidates should be familiar with hypotheses as statements that form the basis of thesis, Coursework assignments. The hypotheses may investigate a geographical concept e.g. ‘A CBD has the highest concentration of comparison shops’. Collecting relevant data, analysis and business life cycle essay, drawing conclusions using the data as evidence can test these.

The width of a river will increase as you move from the thesis source to the mouth The amount of traffic will increase as you move from the rural-urban fringe to the CBD The amount of vegetation will increase as you move inland from the sea (distance = 200metres) The hottest part of the the death of king day will be between 1200 and 1400. Whenever you are doing data collection, the aim is to be as objective as possible. Objective means that no bias or personal opinion affects the outcome of your results. Eur? The opposite to be objective is being subjective. Being subjective simple means that your own personal views and bias has influenced results. It is coursework doctorate programs, up to date (current) You know how the data has been collected i.e. what technique It only includes data that is relevant to thesis eur, your coursework It only covers your study area It is collected in the format that you want. You can study temporal changes e.g. how population has changed over a number of years It can be quicker, especially if the data is on the internet You can study a larger area It may include data that you can not obtain personally e.g. The Death Of King Arthur Essay? salaries. The data may include some personal bias Data collection can be time consuming It can be expensive to travel to places to collect data It is hard to eur, study temporal changes Some data might be unavailable or too dangerous to coursework doctorate programs, collect Only possible to cover a small area.

It is out of date, especially if it has been printed in a book. There might more information than you need The information may include a larger area than your study area You may not know how the thesis data was collected and who collected the data The data might be in the wrong format e.g. in a graph and whale essays, not raw figures. Quantitative data: This is any data that involves figures. Quantitative data is very easy to present and analyse. Eur? Even though it is easy to present it can be very general and exclude some data. Because you are following a pattern you will get better coverage of an area or sample group. Even though you pick the technique, once it is picked, there is business essay, no bias in who gets selected. It is very simple to understand and carry out.

Because you are selecting the systematic technique, there is some bias (subjectivity). Thesis? You decide how often to take a sample. Even with a systematic sample you may end up with an unrepresentative sample e.g. you ask every 10th person to fill in a questionnaire, but every 10th person turns out to be female. Every person or every location/place has a completely equal chance of being selected It is quick and simple. Because it is coursework, totally random, results maybe completely random and not representative e.g. when randomly selecting names out of hat to ask questionnaires too, you only pick females. Stratified sampling: Because both systematic and random sampling can give you a unrepresentative sample, if you have some secondary data that allows you to rank your sample group you can then carry out stratified sampling. Thesis? For example if you are carrying out environmental indexes in the death of king arthur essay, a city that has 12 districts, if you randomly or systematically select four, you may pick th efour best or the four worst.

However, if you know the average income of those 12 areas (census data) you can them rank them 1 to 12 and then randomly or systematically pick a district from thesis eur, each quartile giving you a more representative sample. Decide on your questions (what do you need to whale essays, find out and what type of question are you going to thesis, ask) Whether to do the questions orally or give them in written form (you may have to think about translations) How you are going to, pilot (test) your questions How you are going to record the answers (record them, write them down, trying to remember is eur, unreliable) How you are going to sample (random, systematic, stratified) Remember that you are representing yourself and the school - be polite Remember people don't have to whale essays, answer questions and they may be sensitive about eur, some e.g. age groups and business cycle, income groups. When actually designing your questions you have two real choice; open or closed: You are getting the respondents personal opinion. They are not limited in their response. Some responses might not be relevant to your research Respondents may not understand the question and give you an thesis irrelevant response Results are very hard to analyse using graphs or tables.

All answers will be relevant to your research The results are easy to analyse using graphs and, tables. The results lack personal opinion. Thesis? They can be very generalised You are nearly always forced to have an book college other box meaning you don't know what the respondent thinks Your personal opinion has been placed on the questions (subjective). Specification: Observation – Examples of using observations as an thesis eur enquiry skill to collect data include the recording of crisis, land-use in an urban area or observations of river or coastal features. Maps, recording sheets, field sketches and annotated photographs may all be used to record candidate observations. It is important to always have an other category, because you always find a land use that you have not thought about.

They are more accurate than field sketches They can be good for showing data collection techniques e.g. measuring a river's load They can support data collection findings e.g. they can show an example of a poor environment They can show temporal changes, especially if you can find historical photos. You can annotate and label them. People often include photos that are not relevant e.g. a photo of their friends People forget to label, annotate or refer to thesis, photos, which then makes them irrelevant. People often only photograph the nice things e.g. pretty view and forget the coursework more ugly areas that are just as important e.g. area of thesis eur, pollution They can often contains too much information e.g. The Death? people and thesis, vehicles Because they are two dimensional, depth can be deceptive. Specification: Counts – Pedestrian and traffic counts are two significant examples of this enquiry skill. Appropriate methods for recording the counts should be discussed including the of king arthur layout of recording sheets, instructions and eur, the necessary information required to identify the essays in india sheet following the count (i.e. time, date, location and name of recorder). Geography fieldwork equipment can be used to measure virtually anything, but is most commonly used to take measurements along a river or at eur, a beach. Below are photos and descriptions of some of the most common types of geography equipment. Transect: A transect is basically a line a long which you take measurements.

You may have a transect that runs from the rural-urban fringe to the CBD or a transect that runs from the sea in land through sand dunes. In a real piece of coursework, you would explain how all your data was collected. On Water In India? In your description you would probably contain the following information: Date, time and location of eur, data collection Group size Description and copy of data collection forms used e.g. questionnaires or counts Explanation of whale essays, how the forms were used e.g. Eur? sample size, count period, count technique, etc. Description of business essay, equipment and an explanation of its use.

Instead of writing a methodology in your exam, you may be asked to write a set of instructions, explaining how data collection should be carried out. For example you might be asked to give a set of instructions for doing a traffic count. You might say: Find a safe location near the road your are counting traffic Count the thesis traffic in both directions for a 10 minute period A tally should be used for counting because this is easy and quick At the end of, 10 minutes count up the totals for each type of vehicle. You may also be asked about how your data collection could be improved. Improvements may be made in some of the following ways:

Do counts more regularly e.g. Eur? every one or two hours Do counts, surveys or indexes in more locations Do counts, surveys and indexes on different days of the business cycle week (including weekends) Get two groups doing the thesis eur same survey, index or count so that an average may be taken. Specification: Data presentation techniques: A knowledge of the eur illustrative techniques to present data across the topics for Paper 4 is required. Whale Essays? This should include, various types of graphs, maps and thesis eur, diagrams for example line graphs, bar graphs, divided bar graphs, histograms, flow diagrams, wind rose graphs, isoline maps, scatter graphs, pie graphs, triangular graphs and radial graphs. If you are asked to complete a graph or table, all the data will be there for you so read the data carefully and complete the graph/table/diagram carefully. Shows spatial distribution e.g. dot map Shows variations between regions and countries e.g. choropleth map Visually interesting (interesting colours, symbols) Very bold and clear Easy to understand Clearly shows trends and anomallies. Can disguise intra-region or intra-country variations e.g. Coursework Programs? choropleth map Hard to see trends and anomalies Very complicated to read Symbols take up to much room. Specification: Analysis: Candidates should be able to describe the thesis patterns in data presented in graphs and tables of results. Reference to relevant geographical knowledge and understanding is often required in the interpretation of the doctorate data. Practice of this skill will improve success in Paper 4 questions. Look for trends and correlations (if there is not a overall trend, look for smaller trends) Look for anomalies (things that don't fit the general trend) When ever you refer to trend and anomalies you must support with evidence e.g. facts and figures from graph or table.

Try and explain trends (refer back to theory or other information that you have discovered in your investigation) Try and thesis, explain anomalies. Specification: Formation of, conclusions: Using the thesis evidence from the data, candidates should be able to make judgements on the validity of the original hypothesis or aims of the assignment. Reference is also required of the reliability of the writing book collected data and a critical evaluation of the chosen data collection methods. Refer back to eur, original hypothesis Use some data to support your findings Refer to theory (if mentioned in introduction) - do your findings agree or disagree with theory State what you have learnt from your investigation. Evaluation : In an evaluation you state what went well in your research, but also how it can be improved or extended in the future. If you are asked to write an evaluation, think about the following: What went well (keep this brief) Any problems with data collection e.g. Whale Essays? bad weather, missing data, sampling technique, questions, data collection form Data that could be useful in the future e.g. secondary data from thesis eur, government, more questionnaires (bigger sample) Additional hypothesis that you could have used Problems with time or money that could be changed in the future.

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