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I did not really have that much experience in writing essays . In the past semester I have grown more as a writer than I have in the past years. My essays have. Better , Essay , Essays 1419 Words | 4 Pages. Task D Reflective essay The concepts of cromwell or a villain essay, reflective practice are widely accepted in education and essay 800 words many other . professions, in simple terms, Moon (2004) describes it as a process of 'cognitive housekeeping' whereby a practitioner would explore their own values, beliefs and practice to a professional situation. It is also frequently perceived that a structure is thesis for unemployment rate, helpful to support and encourage more than just simple reflections ,Jones (2009). Popular theorists have emerged in reflective practice. Education , Learning , Lifelong learning 2044 Words | 5 Pages. ? Reflective Essay Jazmine Caballero Walden University Dr. Essay 800 Words. Paul Trautman EDUC 6731: Assessment for was oliver, . Student Learning June 28, 2015 Reflective Essay I can give faith that my assumptions about essay 800 words assessment have been changed throughout this course.
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? University of Technology Jodian Allen I.D #: 1305502 Business Etiquette Assessment Reflective Paper Lecturer: Dr. Jennifer . Essay. Yee-sing Date: September 27, 2013 Reflective Paper Shifting my head slowly from left to cromwell or a villain essay the right I glanced at my classmates, as the essay, lecturer entered the business plan, room I was awfully anxious with muddled thoughts flowing through my head. I sat patiently in the room amongst the others, looking peculiarly sneaky as fear had already numbed my body to a pulp. However. Better , Improve , Learning 814 Words | 3 Pages. ?The Reflective Essay Since joining the class and over the last semester, I have been pursuing English as a subject consequently . Essay Structure. learning the was oliver or a essay, importance of structure 800 words, brainstorming, aspects of organizing prior to writing, revising and persuasive essay on recycling paper collaborating in writing. The subject has provided a gateway through which I have learnt the art of expressing myself, literature, grammar as well as the essay structure, language itself. With Outline. However, this did not come without its challenges. English being my second language, it took me a while to. Clause , Dependent clause , English language 769 Words | 3 Pages. Child Development I (ECE1101) Assessment 1 (Part B) – Reflective Essay By: Muna Farah Student ID: FAR07224027 In this . essay I will be discussing my own developing concept of childhood, my opinion of when childhood starts and ends as well as the importance of childhood.
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At that time, English is the statement for unemployment, most difficult. 2008 singles , Dance-pop songs , Debut albums 970 Words | 3 Pages. Reflective Essay This semester in 1010 has definitely been challenging for me, but I have learned a great deal. Although I’m not . a fan of writing papers, I do have to agree it has helped me increase my knowledge in 800 words, English composition. Before this semester, I had no clue in statement for unemployment, the world what rhetorical skills were, and now I am writing a paper on them. Structure. As a result of English 1010, my ability to think rhetorically has improved greatly. This semester in English 1010, I have learned how to use attributions. Academia , Critical thinking , English-language films 1122 Words | 3 Pages.
Self- reflective Essay In life we go through different developmental stages that lead us to objective certain situations. This has an essay 800 words, . effect on our future direction and which path we choose to take; Whether it be through Education, Career or jobs, family or just shear circumstances, these all contribute and lead us to with certain points in our lives and future endeavors. In this essay I have chosen to write about the major influences that have led me to undertake studying at a tertiary level in 800 words, the course. College , Education , Higher education 1938 Words | 6 Pages. I need in order to present the message I need to convey to my audience in my essays , I did not know how to have sentence variety, and last but . Opportunities. not least I thought it was not an important thing to back up my statements with real examples.
My first writing assignment grade proves that I had a lot to learn my freshman semester at essay structure 800 words, Georgia State University. My first writing assignment for English 1101 was to write an essay about a personal characteristic that made me unique. It sounded like an easy. Creative writing , Essay , Grammar 1197 Words | 3 Pages. rule enforcement---assigned essays on their deleterious, “harmful; injurious; hurtful,” tardiness, a loss of their scheduled spot, and a drop . in their current grade---that being late to standard plan _____’s class is customarily deleteriously unacceptable.
Through the essay 800 words, enforcement of castigations---an essay on tardiness for example---in response to a student’s tardiness, students have learned that lethargy to _____’s class is of this thesis, adversely inexcusable. 800 Words. Any student who has acquired an essay considers himself or herself. Education , Essay , Essays 831 Words | 3 Pages. Danh Nguyen Mr. Goldberg Rhetoric 120 Reflective essay Poverty has always been present in both rich and opportunities essay poor countries . around the world. 800 Words. Vietnam is not an was oliver or a essay, exception. Vietnam, the essay structure 800 words, pearl of the East, shines beautifully with a charming culture and a long glorious history. The Main Of This. However, hiding underneath that beauty, the present of poverty is still out there in the big cities as well as in the countryside. The longing war that Vietnam had been through with Mongolian, China, France, United States and.
Cycle of poverty , Ho Chi Minh , Ho Chi Minh City 1585 Words | 4 Pages. Throughout the summer semester I have had to write a number of essays using a variety of techniques. Before college, I would write assuming . 800 Words. that the only reader, or audience, would be my teacher. Composition helped me realize the many steps that are involved in the writing process from free writing to making a final copy. Essay With Outline. In order to create a good paper you need organize your ideas. Also, I realize what needs to be in a essay and what doesn't. Structure. I have improved in my writing but. Better , Composition studies , Essay 523 Words | 3 Pages. Katrina Wright Reflective Paper February 27, 2010 PDC-111 “Discovering Yourself” The topic “Discovering Yourself” was about the many . different learning styles. This topic also was about how different individuals use various learning styles. Statement Rate. My favorite assignments that I completed when this topic was covered are the Discovery Wheel and the Learning Style Inventory.
The Discovery Wheel weighed strengthens and weaknesses of structure 800 words, self- discovery, career planning, time, reading, notes, and tests. Education , Educational psychology , Learning 888 Words | 3 Pages. Victoria Oslund EDUC 203 12/10/12 Reflective Essay . Persuasive Paper. Teaching and Learning in a Diverse Society has truly opened my eyes to . Essay Structure 800 Words. what I will be facing in a classroom, whether it be the type of students I will be teaching or how I will teach my students a controversial topic. It gave me many different ways to bring up a hard topic that someone people would feel uncomfortably teaching to younger students. Not only did it open my eyes it has helped me think in abstract ways on topics that are new to. Class I railroad , Classroom , Education 2043 Words | 5 Pages.
details once again. It was also somewhat difficult to statement argue what you’re talking about. I want to learn how to write a good Argumentation paper with . good details. The grammar and punctuation don’t get me as bad as the explaining in details in my Essays . Once again I believe most of that comes from essay, distractions at home. Equal Opportunities Essay. If I would have had more time I probably would have found somewhere quiet to type my papers. Essay Structure 800 Words. When there are many of outline, things going on in your life it is a great distraction in. 2004 albums , 2007 singles , Education 615 Words | 2 Pages.
?self-reflection essay I remember first arriving in structure, Haiti. The blast of heat stepping off the plane, cramming onto a mini school bus, and . Persuasive Paper. staring out the structure, windows at the outskirts of Cap Haitian and the rural road to Milot. “I’m in Haiti…,” I kept telling myself, trying to absorb all that I was seeing. The landscape, the thesis statement for unemployment rate, way people live here, the norms of this culture were so different from anything I had seen before. We walked the streets of Milot that first day and I strongly recall the uncomfortable. Cap-Haitien , Haiti , Haitian Creole language 1067 Words | 3 Pages. ?Word count: 1465 Reflective essay on People in Organizations Introduction This is a reflective essay . on my experience of learning “People in organizations” during the essay structure, first half year of my postgraduate study. I will include my personal feelings during the on democracy in pakistan with outline, learning process in teamwork when making the portfolio. Furthermore, I will also include my critical thinking of this module and how I have learnt from it.
Also, there will be an analysis of personal action plan of essay structure 800 words, how I will apply the knowledge. Critical thinking , Human resource management , Human resources 1617 Words | 6 Pages. ? Senior Project Reflective Essay “Intro to was oliver cromwell or a villain Child Care” Being a big sister I have always been introduced into the basics of essay structure, . how to take care a children. For my senior project I chose the self-development option where I took a class at WITC on the main objective of this, “Intro to essay 800 words Child Care.” The class had a Pass/Fail grading system. I decided to take the class as my senior project to the main of this thesis expand my prior knowledge on childcare. Also I took the essay, class to a hero or a villain determine if I want to pursue a career in childcare so I would know what. Child development , Childhood , Day care 842 Words | 4 Pages.
CCCH9009 Protests, Rebellions and Revolutions in Modern China Reflective Essay Name: CHAN Chun Ho, Goofy UID: 3035071486 . Tutorial: Friday 10.30 – 11.20 CCCH 9009 Protests, Rebellions and Revolutions in Modern China Reflective Essay Introduction Ian Johnson’s Wild Grass presents three stories that vividly reflect the essay structure 800 words, social and political problems in the contemporary Chinese regime. Featuring three groups of protagonists, namely the ‘peasant champion’ who filed class-action lawsuit against opportunities essay, the. Chinese people , Communist Party of China , Falun Gong 2051 Words | 6 Pages. Reflective Essay Throughout this course, I have had to 800 words write a number of essays using a variety of techniques. . The concept and cause and effect essays are the two main papers I will use to make an evaluative analysis of the outcomes I have learned, utilizing the writing process in statement for unemployment rate, Basic Writing I, with expectations of receiving the structure 800 words, grade of at the main of this thesis, least a “B” on essay structure, my portfolio. The Main Objective Thesis. This course helped me realize the many steps that are involved in essay structure 800 words, the writing process from equal opportunities essay, gathering ideas to making a final. Abuse , Child abuse , Domestic violence 1027 Words | 3 Pages. Reflective Essay Sample Reference. ? Reflective Essay on to give subcutaneous Injections Introduction This assignment is a reflective account on . nursing skills that I was assigned while on placement in an Organic Mental Health In-Patient Ward, presented with physical conditions and early stages of dementia.
The three skills I want to base my essay on are Subcutaneous Injections, Physiological skill – limited to measurement of essay, blood pressure, Blood Glucose Testing. A brief definition of reflection will be attempted and the importance. Blood glucose monitoring , Hypodermic needle , Insulin 1885 Words | 5 Pages. Reflective essay on communication. the patientaˆ™s purple aˆ?This is meaˆ? folder to try and equal opportunities essay engage them on a more personal level.
With this reflective account I will be . discussing an experience that I had on my ward and 800 words how through reflection I have managed to gain more confidence in my ability to talk to my patients and convey my compassion in the way I communicate with the patients in my care. I will use the reflective framework devised by Atkin and Murphy (1994) to construct this account. It is the objective thesis, right model of essay structure 800 words, reflection. Health care , Health care provider , Illness 1823 Words | 5 Pages. Byrne1 Taylor Byrne MMHS Junior Reflective Essay 24 April 2014 Reflective Essay My name is persuasive on recycling, Taylor, . Essay. I'm a junior here at Murrieta Mesa High School. During the last 3 years i’ve learned a lot and i have been involved with many school activities. Of This Thesis. Being involved on campus is a passion of mine.
I have attended dances, football games and Pep Rally’s. Essay Structure. I have made the effort to attend with the most school spirit. Objective Of This. This year I finally made it into the Leadership club. Now not only essay structure 800 words, can i have the most school. College , Education in the United States , Extracurricular activity 845 Words | 4 Pages. Self Assessment / Reflective Essay.
Assignment #7: Assignment #7: Self Assessment / Reflective Essay Strayer University Abstract In this . Statement For Unemployment. essay I’m sharing with you some of the struggles I endured as a teenager and essay 800 words I explain how I learned from a lot of the with, mistakes I made. Assignment # 7 Self Assessment / Reflective Essay Looking back and reflecting on my life I can now see that I have made many mistakes. Most of the essay 800 words, mistakes I have made were due to equal essay my lack of patience and never listening to older family. Adolescence , Education , Educational psychology 998 Words | 3 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 800 words, the number field, the ideals and units in was oliver cromwell essay, the ring of integers, the extent to 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). 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. Structure! v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ 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. Cromwell Essay! Notations. . . . . . . . . . . . . . . . . . . Structure! . . . . . . Statement For Unemployment Rate! . . 800 Words! . Of This Thesis! . . . . Essay 800 Words! . . . . . . . 5 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objective! . 5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . Essay Structure! . . . . . . . . . 5 Introduction . . . . . . . Opportunities! . . . . . . . . . . . . 800 Words! . . . . . . . . . . . . . Equal Opportunities Essay! . . . . . 1 Exercises . . Essay Structure! . . . . . Persuasive Essay On Recycling Paper! . . . . . . . . . . . . . . . . . . . . . . . Structure 800 Words! . Plan Outline! . . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . 800 Words! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . . On Democracy With Outline! . . . . . . . . . . . . . . . . . . . . . . . 8 Noetherian rings . . . . . Essay Structure 800 Words! . . . . . . . . . . . . Essay On Recycling! . . . . Structure! . . . . . . Essay In Pakistan With! . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . . 800 Words! . . . . . . . . . . . . . . . . 10 Local rings . . . . Persuasive On Recycling! . . . . . . . . . . . . . . . Essay 800 Words! . . Essay In Pakistan! . . . . . Essay 800 Words! . . . . . . . . Persuasive Essay! . . . 10 Rings of fractions . 800 Words! . . . . . . . . . . . . . . . . . . . . . . . . . Objective Of This Thesis! . . . . . . . . 11 The Chinese remainder theorem . . . . . Essay! . . . . . . . . On Democracy With Outline! . . . . . . . . . . . . . 12 Review of tensor products . . . Structure 800 Words! . . . . . . . . . . . . . . . . . . . . A Hero Or A Villain Essay! . . . . . . . 14 Exercise . . . . . . . Essay Structure! . . . . . . . . . . . . . . Was Oliver Cromwell A Hero Or A! . . . . . . . . . . . . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . . . . . . . . . . 19 Dedekind’s proof that the integral elements form a ring . . . . . . . . . Essay Structure 800 Words! . . . . . Equal Opportunities Essay! 20 Integral elements . . . . . . . . . . . . . . . . . . . . Essay! . . . . . Business Plan! . . . . . . . . . 22 Review of bases of A-modules . . . . . . . . . . . . 800 Words! . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . . . Equal! . Essay Structure! . . . . . . . . Essay On Recycling Paper! . . . . . . . . Essay! . . . Essay On Democracy In Pakistan! . . . 25 Review of bilinear forms . . . . 800 Words! . In Pakistan With! . . . . . . . . . . . . . Essay Structure 800 Words! . . . . . . Essay In Pakistan Outline! . . . Essay Structure 800 Words! . . . 26 Discriminants . . . . . . . The Main! . . . Essay Structure 800 Words! . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . . 29 Finding the ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . The Main Of This Thesis! . . . 31 Algorithms for finding the ring of integers . . . . . . . . . . . . . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . 800 Words! . . . . . . . . . . . . . The Main Objective! . . . . . . . Essay! 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . Rate! . . . . . . . . . Essay 800 Words! . Opportunities! . . . . . . 800 Words! . . . . . . . . . Cromwell A Hero Or A! . . . . Essay 800 Words! . 40 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The ideal class group . . . . . . . . . . Thesis Statement! . . . . . . . . . Essay Structure! . . . . . . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . . . . . . . . . Cromwell Or A Villain Essay! . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . Essay Structure! . . . . . . . . . . . . . . . . . . . . Was Oliver Cromwell A Hero Or A Villain Essay! . 51 Modules over Dedekind domains (sketch). Essay 800 Words! . . . . . . . Essay! . . . Structure 800 Words! . . . . . . . . . . . 52.
Factorization in extensions . . . . . . . Objective Of This! . . . . . . . . . . . . . . . Essay! . . . . . . For Unemployment Rate! . 52 The primes that ramify . . . . . . . . . . . Essay! . . . . . . . . . . . . . . . . . . . . 54 Finding factorizations . . . . . . . . Equal Opportunities Essay! . . . . . . . . . . . . . . . Structure! . . Standard Business! . . . . Essay 800 Words! . . . 56 Examples of factorizations . . . . . Essay! . . Essay Structure 800 Words! . . . . . . . . . . . . . . . . . . . Of This Thesis! . . . 57 Eisenstein extensions . . . . . . . . . . . . . . . Essay Structure 800 Words! . . . Essay In Pakistan With! . . . . . . . . . . . . . . 60 Exercises . . . . . . . . . . . . . Essay 800 Words! . . . . . . . Essay On Recycling! . . . Essay 800 Words! . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . Rate! . . . . . . . . . . . . . . . . . . . . . Essay 800 Words! 63 Statement of the main theorem and its consequences . . . . . . . . . . . . . . . . 65 Lattices . . . . . . . . . . . . The Main Of This Thesis! . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay 800 Words! 68 Some calculus . . . . Standard! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Finiteness of the class number . . . . . . . . . . . . . . . Structure 800 Words! . . . . . . . . . . . . 75 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . Was Oliver Or A Villain! . Structure! . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . Standard Business Plan Outline! . . . . . Essay! . . . . . . . . . . . Equal Opportunities Essay! . . . . . Essay Structure! . . . 78. 5 The Unit Theorem 80 Statement of the theorem . Thesis! . . . . . . . . . . . . . . . . Essay Structure 800 Words! . . . . . . . . . . . . . 80 Proof that UK is finitely generated . Objective Of This Thesis! . . . . . . . . . 800 Words! . . . Thesis For Unemployment! . . . . . . . . . . . . 82 Computation of the rank . Essay Structure! . . . . . . . . . . . . . . . . . . . . . . Thesis For Unemployment Rate! . . Essay! . Equal Opportunities Essay! . Structure 800 Words! . . . 83 S -units . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay In Pakistan With! . . . . . . . . . . . . . 85 Example: CM fields . . . . . . . . . . . . . . . . . . . . . . . . . Essay! . Persuasive Essay On Recycling! . . . . . . . 86 Example: real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . Essay Structure! . . . . . . . . . . . . . . . 87 Finding .K/ . Of This! . . . . . . . . . . . . . . . . Essay Structure! . . . . Persuasive Essay Paper! . . . . . . . . Essay 800 Words! . Essay Paper! . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . Essay Structure 800 Words! . . . . . Equal Opportunities Essay! . . . . . . . . 89 Regulators . . . . . . Essay 800 Words! . . . . . . . . . . . Cromwell A Hero Or A Villain Essay! . . . . . . . Essay Structure 800 Words! . The Main Thesis! . . . . . . . . . . . . . 89 Exercises . . . . . . . . . . . . . . . . . . . . 800 Words! . . Objective Of This Thesis! . . . . Essay! . . A Hero Or A! . . . . . . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure 800 Words! . . . 91 Class numbers of cyclotomic fields . . . . Paper! . . . . . . . . . . . . . . . . . . . . Essay! . 97 Units in cyclotomic fields . . . . . The Main Thesis! . . 800 Words! . . . . . . . . . . . . . . . . . . . . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . . . 98 Exercises . . . . . . . . . . . . . . . Of This! . Structure! . Of This! . . . . . . . . . . . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . . 800 Words! . . . . . Essay In Pakistan With! . . . . . . . . . . . . . . . . . . . . . Structure! . . . . . . . . 101 Nonarchimedean valuations . . . Or A Essay! . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . Essay Structure 800 Words! . . . . . . . . . . . . 103 Properties of discrete valuations . . . . Essay On Democracy Outline! . . . Essay Structure 800 Words! . . . . . . . . . . . . . . . . . . . 105 Complete list of valuations for persuasive on recycling the rational numbers . . . . . . . . . . . Structure! . . Of This! . . . 105 The primes of a number field . . . . . . . . . . . . . . . Structure! . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . The Main Objective! . . . . . . . . . . . . . Structure 800 Words! . . . . . 109 Completions . . . Standard Business Plan Outline! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 Words! . . . . . 110 Completions in the nonarchimedean case . . . . . . . Persuasive Essay On Recycling! . . . . . . . . . . . . . . . 111 Newton’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure 800 Words! . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . . Essay With Outline! . . . . . . 118.
Newton’s polygon . . . . . . . . . . . . . . . . . . . . Essay Structure! . . . . . . . . . . . . . . 120 Locally compact fields . . . . . . . . . . . . On Recycling Paper! . Essay Structure 800 Words! . . . . . Thesis Rate! . . . . . . . . . . 800 Words! . . . 122 Unramified extensions of a local field . Equal! . . . . . Structure! . . . . . Thesis Rate! . . . . . . . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . Structure! . . . Opportunities Essay! . . . . . . . . 125 Ramification groups . . . . . . . . . . . Essay Structure 800 Words! . . . Thesis Statement! . . . . . . . . Essay Structure! . Was Oliver Cromwell Villain! . . . . . . . . . . 126 Krasner’s lemma and applications . . . . . . . . . . . . . . . Structure! . . . . . . . . . . 127 Exercises . . . . . . . . . Standard Business Plan Outline! . . . . . . . . . . . . . . . Structure 800 Words! . . . . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . For Unemployment Rate! . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay 800 Words! . . 131 The product formula . . . . . . . . . . . . . . . . . . . . . . . . Was Oliver Cromwell Or A Villain! . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay 800 Words! . Persuasive On Recycling Paper! . . . 137 Examples . . . . . . . . . . . . . . . Structure 800 Words! . . . . . . . . . . . . . . . . . . . . . . . Persuasive Essay Paper! 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . Essay Structure! . . . Was Oliver! . . . . Essay 800 Words! . . The Main Of This! . . . . 146 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay Structure! . . . . . . . . . . 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 was oliver villain a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . Essay! The empty set is denoted by ;. The cardinality of a set S is denoted by 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 standard, 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 definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to structure 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 the main objective thesis 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 www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011. GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the 800 words following for was oliver a hero villain essay 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- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. 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 problem, but neglected to prove that the algorithm always works. EULER (1707–1783). He introduced analysis into 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 structure 800 words 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 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 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). Cromwell Or A Villain! He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for structure 800 words 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. Essay Paper! EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and 800 words 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 generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).
RIEMANN (1826–1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. On Recycling! 1The Indian mathematician Bhaskara (12th century) knew general rules for 800 words 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 ring of integers in a number field, by proving that ideals factor uniquely into products of statement for unemployment, prime ideals in such rings, and by essay, showing that, modulo principal ideals, they fall into 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).
He gave the essay on recycling first definition of the field of 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). He proved the essay structure 800 words fundamental theorems of abelian class field theory, as conjectured by Weber and 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 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 standard business Hasse (local-global) principle for all quadratic forms over essay structure number fields, and contributed to the main objective of this thesis the classification of central simple algebras over number fields. Essay Structure! BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of 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– ). Standard Business Plan! He proved new results in group cohomology, which allowed him to give an elegant reformulation of 800 words, 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. Plan Outline! 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 a given number, does not also belong to 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 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. 800 Words! 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 a hero or a villain, 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 order to fully understand the arithmetic of essay, K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an persuasive essay on recycling 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 essay 800 words equation with coefficients a1; : : : ;an in cromwell, Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. Essay Structure! We shall show (2.11) that ? is an algebraic integer if and essay in pakistan outline 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. Essay Structure 800 Words! .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 essay on recycling paper 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 800 words, 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 was oliver cromwell a hero (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of 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 structure uniqueness that fails. Thesis! 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 essay 800 words 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. In fact, in an integral domain in which factorizations exist (e.g. Thesis Statement! a Noetherian ring), factorization is 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 essay 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 “prime numbers” so that, for equal example, in structure 800 words, 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. 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 objective all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an essay 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 standard plan outline 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. .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 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 structure 800 words, ideals such that every ideal is of the form a .a/ for thesis for unemployment rate 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 essay 800 words 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. Thesis Statement! 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 essay 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 essay in pakistan with 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 essay structure 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 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 cromwell or a essay 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?. 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. 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 Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in 800 words, 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 statement for unemployment, 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 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 essay structure 800 words, the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the essay in pakistan ring of integers in Q? p d? is essay 800 words as described. 0-2 Complete the 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. Outline! We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . Essay Structure 800 Words! ;?m of in pakistan with outline, B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of essay 800 words, all polynomials in the ?i with coefficients in A, i.e., elements of the form X. ai1. Persuasive Essay Paper! 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; . ;?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 ai . It consists of essay 800 words, finite sums. 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. 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; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an statement for unemployment 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 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. 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 be prime if ./ is a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is structure maximal if it is standard business plan 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 structure 800 words, 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 products of rings. PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and on democracy in pakistan 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 essay 800 words, 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. On Democracy With Outline! Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an 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 essay 800 words form a1 am with ai an ideal in equal essay, Ai ; moreover, a1 am is prime if and only if there is 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 essay structure 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 on democracy with outline hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the essay ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . Essay On Democracy In Pakistan Outline! 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.. From (b) we know that this process will lead to a maximal element after only 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 zero ideal, and structure 800 words 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 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 is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of objective, irreducible elements, it is not itself irreducible, and 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 essay, .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an essay irreducible element of 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 essay structure, 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. 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 in pakistan with M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and 800 words 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 persuasive, 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 claim shows that the chain itself becomes constant. 2. 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 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. Essay! 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 in pakistan with outline, 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 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 essay structure 800 words 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 business plan outline 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 essay 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 ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an integral domainA and objective thesis a multiplicative subset S ofA. For an ideal a of structure 800 words, 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 with so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . 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 S?1A; the inverse map is p 7! pA. PROOF. It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in structure, 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 persuasive on recycling paper, 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 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 800 words, ideals. Integersm and n are relatively prime if and statement only if .m;n/D Z, i.e., if and essay structure 800 words only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. Persuasive On Recycling! of the mi . Thus T .mi / . Q mi /, which equals. Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the 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 essay structure congruences. Objective Thesis! 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. Essay Structure 800 Words! As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. 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 i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so. We can now apply the theorem in the case nD 2 to equal opportunities essay obtain an essay structure 800 words element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. Opportunities! 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 essay structure 800 words 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. 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 on democracy in pakistan with outline 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 800 words, 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 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 is unique (up to persuasive on recycling a unique isomorphism making the obvious diagram commute). We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is essay 800 words 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 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 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 n respectively, then V ?kW is a vector space over opportunities k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Essay Structure! Then. Thesis Statement! .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 mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. Structure 800 Words! : : : . 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 thesis rate 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 structure 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. Thesis Statement For Unemployment Rate! 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 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 essay structure 800 words isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.
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 of this thesis B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for essay 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 a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. 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. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is on recycling 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 essay 800 words some ? 2K. Let f .X/ be the 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). Because K is standard business plan separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. Essay 800 Words! that are relatively prime in equal opportunities, pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .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 structure a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of essay on democracy with outline, ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for essay ? 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 was oliver cromwell a hero, the i th factor is. We note that it is essential to assume in (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 essay, Dedekind for rings of algebraic integers, and later by others in polynomial rings. 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 standard business Emmy Noether). Embed this document on your website. If you don't receive any email, please check your Junk Mail box.
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Something That Matters by Blake Mycoskie, I believed that a person should only equal set fairly easy and attainable goals for themselves. Now my attitude is that . you should pursue what makes you happy, and you shouldn’t let struggles stop you. When I read examples about following a passion, overcoming a fear, and being persistent in your efforts, my prior opinion began to change. Structure? In the for unemployment rate book Blake first describes his journey with creating and growing TOMS. Blake was on a vacation trip in essay, Argentina when. 1083 Words | 3 Pages.
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A Great Way to essay 800 words Care , Cannon , Columbine High School massacre 1438 Words | 4 Pages. ? Essay Option #2 If you don't want to take school seriously, then I guess I’ll look forward to seeing you working your shift at McDonald’s . when I stop by for the fries”. This is what my mom told me when I was an eighth grader. My parents are very accomplished people; my mother is a registered nurse and my father is a lawyer. So her words cut deep and hit me where it hurt. Why was she telling me this? Because I deserved every bit of it. I was at a point in my life when I was very immature. I was. College , Father , High school 1107 Words | 4 Pages.
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Examples of Moral Essays Teenagers in Malaysia. Web Images Maps News Translate Books Gmail more ¡ Help | Sign in essay, Translate too easy to believe the other. For example , if they felt their . friends closer and essay on democracy in pakistan with more understand, they can easily be in accordance with what is essay structure, done by friends. They forget that the friends sometimes can not eat friends. In this situation many young adults fall into the social ills. Influence of Mass Media The mass media also contribute to the occurrence of social problems among young adults today.
Not everything. Adolescence , Decadence , Decline 1562 Words | 6 Pages. TASK 4: Assessed Essay WRITING TASK 4 is your assessed essay and you will need to write an essay of 1200-1400 . words. This is the same length as the essay in WRITING TASK 3. WRITING TASK 4 will form 80% of persuasive on recycling your mark for structure, writing on the Pre-Sessional course. Please select a title from the list here: 1. Discuss the impact a particular theory has had on your area of study in terms of application, use and limitations. 2. Explain the equal opportunities essay problems caused by structure offshore oil spills, giving an example of a particular.
Citation , Essay , Marketing 1150 Words | 4 Pages. social—have their own languages, rituals, and symbols which can be understood and manipulated by on democracy in pakistan with someone who is culturally literate. Second, being literate . can mean having knowledge or competence. For example , we speak of people being computer literate or politically literate. For your first essay , try to focus on a moment or a period in your life when you realized the significance of being literate in essay 800 words, this fashion. Did you have trouble using a computer to equal opportunities essay register for classes? Did you fit into. Essay , Knowledge , Literacy 1120 Words | 4 Pages.
people the way I wanted to be treated, for example I acted aggressive towards them and I miss understood them each time that they wanted to . Essay? help. In this essay I will give two ways/solution on how my attitude will change towards people around me. First I will act less aggressive towards them and secondly I will try to understand if maybe they are trying to help me. I should be less aggressive towards people in class and people around us, for example last week on Friday during economics I was. Aggression , Anger , Debut albums 517 Words | 2 Pages.