Abstract algebra – Topics for talks
Instructor: M. Grützmann (古梅西)
Textbook: P.A. Grillet, Abstract Algebra, GTM, Springer (2007), ISBN 978-0-387-71568-1. (Here is an online version.1)
Course home page: www.grutzmann.de/algebra.
I. Groups (5+ weeks)
1.4. Free groups (自由群), free products (自由积), and presentations (群的展示)
- lemma: existence of free groups (universal property),
- thm: existence and uniqueness of groups from generators and relations,
- examples: dihedral group, unit quaternions.
- defn+prop: free product (universal property),
- outlook*: amalgamated product (混合积)
1.5 Direct products (真积) － 王陆华
- Direct products with examples
- criterion for G≅ G1×G2
- structure theorem of finitely generated Abelian groups (with idea of proof: Torsion part, free part)
1.6 The Krull–Schmidt theorem (克鲁尔–施密特定理) – 张阅化
- The K-S thm with the idea of proof
1.8 Structure of symmetric groups (置换群) – 杨若松
- Every (finite) group isomorphic to a subgroup of a (finite) permutation group
- Definition of Sn and proof of group structure
- map and orbit notation of permutations, product of non-disjoint orbits
- signum of a permutation and An
- prop: An is simple for n≥5.
1.10 Small groups (分类的小群) – 曹宁
- Cn, Sn, An,Dn
- groups of order 2p where p a prime,
- groups of oder 8 and 12,
- All groups up to order 15,
- as time permits: more examples of structure proofs |G|=27, |G|=20, |G|=30
- what happens for |G|=60 ?
1.11 The general linear group (一般线性群) – 朱翠影
- review determinant and adjoint formula (for A-1)
- GL(V), GLn(F)
- PSLn(F) is simple for n≥3 (idea of proof optional).
- O(n), SO(n),
- U(n), SU(n),
- any other three continuous (or matrix) groups, e.g. ISO(Rn)=O(n)⋉Rn ⊂ GLn+1(R), Torus Tn= (S1)n ⊂ U(n), Spn(F), upper triangular matrices (with or without diagonal), …
II. Rings and algebras (换理论与代数, 3 weeks)
2.2 Homomorphisms (同态), subrings (子环), and ideals (理想) – 荆慧双
- quotient ring
- sum, intersection, and product of ideals
- isomorphism theorems (for rings, proofs are homework)
2.3 Domains (整环) and fields (域) – 代菡
- Definitions, + a|b,
- Examples: integers, polynomials over a domain,
- field of fractions,
- definition: unit, irreducible element, maximal ideal, and prime ideal;
- examples and counter-examples for relations between them
2.4 Principal ideal domains (主理想环) – 刘意
- Examples: integers, Polynomials in one variable
- gcd and Euclidian algorithm (辗转相除法, Euclidian domain)
- Prop: proper prime ideals are generated by irreducible elements, these are maximal
- A.C.C. (升链条件)
2.5 Unique factorization domains (唯一分解整环) – 郝志香
- Proposition: PIDs are UFDs
- Examples: Integers, Polynomials in one variable
- Eisenstein's criterion of irreducibility (艾森斯坦判别法)
- Example: cyclotomic polynomials (分圆多项式)
III Field extensions (域扩张) and Galois theory (伽罗瓦理论, 5 weeks)
3.3 Separable extensions (可分扩张) – 王会菊
- Lemma: characteristic 0 or algebraic over Fp is separable
- Counter example: F2(ξ,η)/(η2-ξ)
- Lemma: Inter fields remain separable
3.4 Resultants (结式) and discriminants (判别式) – 周Rui
- solution formula for quadratic polynomials, discriminant;
- solution strategy of cubic polynomials, discriminant;
- general definition of discriminant, example: degree 4;
- apparent problem: No solution formula for quintic (or higher) polynomial equations;
- alternate recipies to decide whether a polynom has multiple roots (discriminant) or two polynomials common roots (resultant) – derivative and Euclidean algorithm.
3.6 Galois extensions (伽罗瓦扩张) and the correspondence principle (对应原理) – Cheng明宏
- Lemma: Q and Fp have no automorphisms
- Example: automorphism of C
- Definition: Galois extension, Galois group
- Theorem: correspondence principle (both ways), normal subgroups
- idea of proof
- example: subfield scheme of Q(√2,√3)
3.8 Cyclotomic (分圆), cyclic extensions (循环扩张) and Solvability by radicals (可解用根式) – 武乐云
- Theorem: An extension factors into a cyclotomic tower iff its Galois group is solvable (with idea of proof)
- Example of non-solvable
3.10 Geometric constructions (尺规作图) – 郭欠霞
- Theorem: Constructability of rad2Q with ruler and compass (with idea of proof)
- Counter-Examples: doubling the cube, trisecting the angle
- Constructability of regular n-gons (正多边形)
3.12 Outlook: Algebraic geometry (代数几何) – 何挺
- Definition: algebraic sets / varieties, coordinate ring, dimension
- irreducible component, radical ideal with examples
- algebraic morphism, rational map, isomorphisms with examples
- Algebra, 2nd edition, Addison Wesley (2010), ISBN 978-013-241-377-0.
- Lectures in abstract algebra: 1 basic concepts, 3 Fields and Galois theory, Springer, (2000), ISBN 750-620-060-0, 750-620-062-7.
- 抽象代数基础 (Basic Algebra), Springer (2007)/ Higher Education Press, ISBN 978-730-214-407-6.
- A first course in abstract algebra, 3rd edition, Prentice Hall (2006), ISBN 978-711-121-262-1.
- 近世代数 (International Algebra), 3rd edition, Higher Education Press (2011), ISBN 978-704-030-072-7.
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