Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
Leopold Kronecker
Pasha Zusmanovich

6ALGS Algebraic Structures, summer semester 2023/2024

SYLLABUS
LITERATURE (see here for the precise titles and further info)
Main: Additional:
A brief presentation covering some of the topics: pdf TeX
Group Explorer


A BRIEF SYNOPSIS
(each class lasted approximately 3 hours, inless specified otherwise)

Class 1 February 12, 2024:
Organizational issues. General notion of an algebraic system. The idea of symmetries. Symmetries of an equilateral triangle form the group \(S_3\). Definition of a group, examples of groups (permutations, matrices). Subgroup. Abelian groups. The inverse of an element in a group is unique, the unit in a group is unique. Groups of order 2. Galois and Abel.
(Slides, pp.3-8; Lang, pp.7-9; Mac Lane-Birkhoff, pp.43-48,50,63-64; Carter, pp.3-21,25-40,45,48-51,78-80; Shafarevich, pp.96-102,108,110; Vinberg, pp.139-140).

Class 2 February 19, 2024:
Cyclic groups. Order of an element of a group. Groups of order 3. Homomorphism, isomorphism of groups, examples. Example of two non-isomorphic groups of order 4.
(Slides, pp.7,10-11,18; Lang, pp.8-10; Mac Lane-Birkhoff, p.44; Carter, p.159-163; Shafarevich, pp.104-105; Vinberg, pp.163-165).

Class 3 February 26, 2024:
Discussion of homeworks 1-2. Center of a group, examples. The special linear group. Direct product of groups, center of the direct product.
(Slides, p.17; Lang, p.9; Carter, pp.117-128).

Class 4 March 4, 2024:
Discussion of homeworks 3-6. Automorphism group, examples. Inner automorphisms. Normal subgroup.
(Slides, p.11-12; Lang, pp.13-14,26; Carter, pp.132-134; Shafarevich, pp.105-106; Vinberg, pp.161-162).

Class 5 March 18, 2024:
Discussion of homeworks 7-8. Cosets. Lagrange's theorem and its consequences. Quotients. Simple groups.
(Slides, pp.12-14,24; Lang, pp.12-13,26; Mac Lane-Birkhoff, pp.72-74,79; Carter, pp.102-108,132-139,163-167; Shafarevich, pp.105-106,109; Vinberg, pp.155-158,161-162).

Class 6 March 25, 2024:
Discussion of homeworks 9-11. The group of inner automorphisms is a normal subgroup of the group of all automorphisms. Kernel. The first, second and third homomorphism theorems.
(Slides, pp.15-16; Lang, pp.11,16-17; Mac Lane-Birkhoff, pp.75-77,79,411; Carter, pp.163-169; Shafarevich, pp.106-107; Vinberg, pp.165-167).

Class 7 April 8, 2024:
Discussion of homeworks 12-13. Rings, fields, algebras. Examples: number fields, matrices, polynomials, GF(2), GF(3). Homomorphism and isomorphism of rings. Rings consisting of 1,2,3 elements, algebras of dimension 1.
(Slides, pp.26-27,33-35,41-42; Lang, pp.83-84,86; Mac Lane-Birkhoff, pp.85-87; Shafarevich, pp.17-18,62-63; Vinberg, pp.7-9,27-32; note that both Lang and Mac Lane-Birkhoff assume that rings have a unit).
Additional read: B. Poonen, Why all rings should have a 1, Math. Magazine 92 (2019), no.1, 58-62.

Class 8 April 15, 2024 (1h 20min):
Discussion of homeworks 14-17.

Class 9 April 22, 2024:
Discussion of homeworks 18-19. Subrings and subalgebras. Ideals and quotiens, examples. Principal ideals of a commutative ring. Simple rings and algebras. The matrix algebra \(M_n(K)\) is simple. The first isomorphism theorem.
(Slides, pp.29-30, Lang, pp.86-89; Mac Lane-Birkhoff, pp.95-98; Shafarevich, pp.26,28-29; Vinberg, pp.11-12)

Class 10 April 29, 2024:
Discussion of homework 20. The second and third homomorphism theorems for rings and algebras. Quaternions. Tensor product of algebras.
Simplicity of the quaternion algebra: pdf TeX
(Slides, p.44,48-50; Mac Lane-Birkhoff, pp.281-283,319-325; Shafarevich, pp.65-66; Vinberg, pp.295-300,459-460).

Class 11 May 6, 2024 (2h30min):
Characteristic of a field. Prime subfields. Finite fields. Construction of GF(4). Algebraic and transendental extensions of fields. Algebraically closed fields, algebraic closure.
(Lang, pp.223-224,244-247; Mac Lane-Birkhoff, pp.120-121,281-283; Shafarevich, pp.65-66; Vinberg, pp.459-460).


HOMEWORKS

EXAM

Set 1: pdf TeX   Set 2: pdf TeX   Set 3: pdf TeX   Set 4: pdf TeX   Set 5: pdf TeX   Set 6: pdf TeX   Set 8: pdf TeX


A brief synopsis, videos, and homeworks from this course at the previous semesters


Created: Tue Feb 23 2021
Last modified: Mon Jun 17 14:39:29 CEST 2024