추상대수학

개요

• 현대대수학의 기본적인 언어이자 대상인, 군, 환, 체의 기본적인 용어 및 이론을 공부함.
• 갈루아 이론 - 군론을 통해 확장체 혹은 대수방정식의 해가 가진 대칭성을 들여다 봄.

선수 과목 또는 알고 있으면 좋은 것들

• 고교 수준의 대수학
  • 다항식, 다항방정식
• 기초적인 선형대수학
  • 기저, 차원, 선형사상, 행렬, 행렬식

다루는 대상

• 군(group)
  • 대칭성을 기술하는 언어
  • 항등원, 역원,
• 환(ring)
  • 덧셈, 뺄셈, 곱하기가 가능하며, 덧셈과 곱셈 사이에 분배법칙이 성립.
  • 정수의 집합, 다항식의 집합, n x n 행렬들의 집합
• 체(field)
  • 실수, 복소수와 같이 사칙연산이 가능.
  • 좀더 일반적으로 곱셈의 교환법칙을 가정하지 않는 경우는 division ring이라 부름.

중요한 개념 및 정리

• 순환군
• 군론
• 유한생성 아벨군의 기본정리
• 체론(field theory)
• ideal
• 유한체
• 갈루아 체확장

유명한 정리 혹은 생각할만한 문제

• 대수학의 기본정리(The fundamental theorem of algebras)의 대수적 증명은 가능한가?
• 해밀턴의 사원수
  • 아래 참고할만한 자료
    • The Impossibility of a Division Algebra of Vectors in Three Dimensional Space
    • Hamilton's Discovery of Quaternions
• 가우스와 정17각형의 작도
• 그리스 3대 작도 불가능문제를 군론을 통해 해결할 수 있음.
• 일반적인 5차 이상의 방정식의 대수적 해가 존재하지 않음에 대한 아벨의 증명
• 유클리드 도메인이 아닌 PID
  • 아래 참고할만한 자료
    • A Principal Ideal Domain That Is Not a Euclidean Domain
• 7개의 프리즈 패턴
• 17 Plane Crystallographic groups

다른 과목과의 관련성

• 정수론
• 선형대수학
• 대수곡선론
  • 대수기하학 입문으로서의 대수곡선론
• 대수적위상수학
  • 군론
    • fundamental group을 정의하기 위해 필요
    • covering space의 deck transformation group
  • 유한생성 아벨군의 기본정리
    • 호몰로지를 이해하기 위해 필요
• 조합론
  • 번사이드 보조정리

관련된 대학원 과목 또는 더 공부하면 좋은 것들

• 펠릭스 클라인의 '정이십면체와 5차방정식'
• semisimple rings
  • 아틴-웨더번 정리(Artin–Wedderburn theorem)
• 유한군의 표현론
• 대수적정수론
• Classical groups

매스매티카 파일 및 계산 리소스

• http://homepage.math.uiowa.edu/~goodman/22m121.dir/2006/22m121.html

표준적인 교과서

• John B. Fraleigh A First Course in Abstract Algebra

관련도서

• Israel Kleiner A History of Abstract Algebra

관련논문

• I. G. Bashmakova and A. N. Rudakov The Evolution of Algebra 1800-1870 , The American Mathematical Monthly, Vol. 102, No. 3 (Mar., 1995), pp. 266-270
• The Evolution of Group Theory: A Brief Survey
  • Israel Kleiner, Mathematics Magazine, Vol. 59, No. 4 (Oct., 1986), pp. 195-215
• A History of Lagrange's Theorem on Groups
  • Richard L. Roth, Mathematics Magazine, Vol. 74, No. 2 (Apr., 2001), pp. 99-108
• Hamilton's Discovery of Quaternions
  • B. L. van der Waerden, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 227-234
• The Genesis of the Abstract Ring Concept
  • Israel Kleiner, The American Mathematical Monthly, Vol. 103, No. 5 (May, 1996), pp. 417-424
• A Historically Focused Course in Abstract Algebra
  • Israel Kleiner, Mathematics Magazine, Vol. 71, No. 2 (Apr., 1998), pp. 105-111
• Galois Theory for Beginners
  • John Stillwell, The American Mathematical Monthly, Vol. 101, No. 1 (Jan., 1994), pp. 22-27
• Niels Hendrik Abel and Equations of the Fifth Degree
  • Michael I. Rosen, The American Mathematical Monthly, Vol. 102, No. 6 (Jun. - Jul., 1995), pp. 495-505
• A Principal Ideal Domain That Is Not a Euclidean Domain
  • Oscar A. Campoli, The American Mathematical Monthly, Vol. 95, No. 9 (Nov., 1988), pp. 868-871
• Principal Ideal Domains are Almost Euclidean
  • John Greene, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 154-156

노트

위키데이터

• ID : Q159943

말뭉치

1. Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner.
2. Five of them will be taking college courses in differential equations, abstract algebra and discrete mathematics as 10th-graders at Pasadena High School this fall.
3. At the height of her mathematical powers, doing new work on abstract algebra, Noether died after an operation on an ovarian cyst.
4. If the operations satisfy familiar arithmetic rules (such as associativity, commutativity, and distributivity) the set will have a particularly “rich” algebraic structure.
5. Sets with the richest algebraic structure are known as fields.
6. In fact, finite fields motivated the early development of abstract algebra.
7. Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems.
8. discrete mathematics are sometimes considered branches of abstract algebra.
9. In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
10. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.
11. Algebraic structures, with their associated homomorphisms, form mathematical categories.
12. As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra.
13. But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior!
14. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra.
15. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.
16. Algebra has also played a significant role in clarifying and highlighting notions of logic, at the core of exact philosophy for millennia.
17. A number of branches of mathematics have found algebra such an effective tool that they have spawned algebraic subbranches.
18. Groups, rings and fields only scratch the surface of abstract algebra.
19. Boolean algebras abstract the algebra of sets.
20. "Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra.
21. The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra.
22. This book is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more.
23. Primarily, we will follow Wikibooks' Abstract Algebra textbook.
24. A good textbook to pick up is Topics in Algebra by I.N. Herstein.
25. Shed the societal and cultural narratives holding you back and let step-by-step A First Course in Abstract Algebra textbook solutions reorient your old paradigms.
26. Unlock your A First Course in Abstract Algebra PDF (Profound Dynamic Fulfillment) today.
27. This course develops in the theme of "Arithmetic congruence, and abstract algebraic structures.
28. This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences.
29. Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world.
30. Perhaps no other subject of undergraduate mathematics is as challenging to learn (and to teach) as abstract algebra.
31. Many of the applications of modern algebra are explained in a thoughtful way that will help motivate students to study abstract concepts.
32. Abstract Algebra by Dummit & Foote is a standard textbook used by colleges and universities.
33. It covers all the topics for a solid first course in Abstract Algebra.
34. Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication.
35. Noether went into research and more or less invented the field of abstract algebra.
36. The main aim of the course is to introduce you to basic concepts from abstract algebra, especially the notion of a group.
37. The course will help prepare you for further study in abstract algebra as well as familiarize you with tools essential in many other areas of mathematics.
38. Computational Problems in Abstract Algebra provides information pertinent to the application of computers to abstract algebra.
39. The final chapter deals with the computational problems related to invariant factors in linear algebra.
40. Mathematicians as well as students of algebra will find this book useful.

메타데이터

위키데이터

• ID : Q159943

Spacy 패턴 목록

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• [{'LOWER': 'modern'}, {'LEMMA': 'algebra'}]
• [{'LEMMA': 'algebra'}]
• [{'LEMMA': 'algebraic'}]