# 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)

## I. Groups (5+ weeks)

### 1.4. Free groups (自由群), free products (自由积), and presentations (群的展示)

• definitions,
• 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
• Examples

### 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)
• SLn(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

• Definition,
• Examples
• 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

• Definition,
• 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

• Definition
• 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

• Definition
• 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

• Definition
• Example
• 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
• Examples
• irreducible component, radical ideal with examples
• algebraic morphism, rational map, isomorphisms with examples

### Further literature

1. : Algebra, 2nd edition, Addison Wesley (2010), ISBN 978-013-241-377-0.
2. : Lectures in abstract algebra: 1 basic concepts, 3 Fields and Galois theory, Springer, (2000), ISBN 750-620-060-0, 750-620-062-7.
3. Kezheng): 抽象代数基础 (Basic Algebra), Springer (2007)/ Higher Education Press, ISBN 978-730-214-407-6.
4. : A first course in abstract algebra, 3rd edition, Prentice Hall (2006), ISBN 978-711-121-262-1.
5. Zixu): 近世代数 (International Algebra), 3rd edition, Higher Education Press (2011), ISBN 978-704-030-072-7.

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