Math 21-373 - Algebraic Structures
Sevak Mkrtchyan
7121 Wean Hall
e-mail: sevakm (at_sign) andrew.cmu.edu

Course Description

We will study groups, rings, fields and Galois theory. Time permitting some representation theory.

Lectures

MWF 9:30-10:20 in WH 5409

Office Hours:

M 10:30-11:20, Th 9:30-10:20AM (7121 Wean Hall)

Textbook

The textbook for the course is: Herstein, Topics in Algebra, 2nd edition

Grades

Your grade in the class will be based on the following weights:
25% - Homework assignments
20% - Midterm exam 1 (Monday, September 30, in class)
20% - Midterm exam 2 (Monday, November 4, in class)
35% - Final exam

Homework

Homework assignments will be posted online and collected in class at the beginning of class on Fridays. Homework is due by 5pm on the due date in my office. No late homework will be accepted. The two lowest homework grades will be dropped.

Although I discourage it, you are allowed to collaborate on homework assignments. However, writing up of the solutions should be done individually. You should never share written down solutions, which will be submitted as homework, with other students.

This is a schedule for what I have covered and what I plan to cover in class each day. This section will be updated regularly during the semester.

Date
 Topic
 Homework
 Due
Mon, August 26 Introduction to group theory. 1.2.8,Extra credit 1.2.6 Aug 30
Wed, August 28 Examples of groups. Dihedral groups. 2.3: 1a,3,8,11,19 Aug 30
Fri, August 30 Symmetric groups. Fields. 2.3: 4,5,10,14,16,17,21, Extra credit 2.3.26 Sep 6
Mon, September 2 Labor day; No classes
Wed, September 4 Matrix groups.
Fri, September 6 Subgroups. Lagrange's theorem. Homework 3 Sep 13
Mon, September 9 Homomorphisms.
Wed, September 11 Normal subgroups. 2.6: 5,10,21 Sep 20
Fri, September 13 The first isomorphism theorem. 2.7: 2,5,8,10 Sep 20
Mon, September 16 Simple groups.
Wed, September 18 The second and third isomorphism theorems.
Fri, September 20 The fourth isomorphism theorem. Homework 5 Sep 27
Mon, September 23 Projective special linear groups.
Wed, September 25 Automorphisms
Fri, September 27 Cayley's theorem 2.8: 1(No need to justify),5,19,20,21 Oct 4
Mon, September 30 Midterm 1
Wed, October 2 Permutation groups 2.9: 5,6,7 Oct 11
Fri, October 4 Permutation groups 2.10: 2,3,6,11,14 Oct 11
Mon, October 7 The class equation 2.10: 21,22 Oct 21
Wed, October 9 Group actions
Fri, October 11 Cauchy's theorem 2.11: 5,11 Oct 21
Mon, October 14 Sylow's theorem 2.12: 7,9,13 Oct 21
Wed, October 16 Direct products 2.13: 5,6,7,8 Oct 25
Fri, October 18 Mid-Semester Break; No Classes
Mon, October 21 Classification of finite abelian groups 2.14: 3,9 Oct 25
Wed, October 23 Classification of finite abelian groups
Fri, October 25 Classification of finite abelian groups Homework 10 Nov 1
Mon, October 28 Rings, integral domains. 3.2:6,7,8 Nov 15
Wed, October 30 Homomorphisms of rings, ideals. Isomorphism theorems. 3.4:1,2,5,6,20 Nov 15
Fri, November 1 Maximal ideals. Field of fractions of an integral domain. 3.6:4,5(extra credit) Nov 15
Mon, November 4 Midterm 2
Wed, November 6 Euclidean rings. 3.7:3, Read Sections 3.8 and 3.9 Nov 15
Fri, November 8 Unique factorization into primes. Ring of polynomials. 3.9:3,5 Nov 22
Mon, November 11 Ring of polynomials over a UFD is a UFD. 3.10:3 Nov 22
Wed, November 13 Finite field extensions. 3.11:4,8,14 Nov 22
Fri, November 15 Splitting fields 5.1:4,8,9 Nov 22
Mon, November 18 Introduction to Galois theory 5.3:6ac,10ab,11,14 Dec 6
Wed, November 20 Galois extensions 5.5:6,10 Dec 6
Fri, November 22 Symmetric rational functions 5.6:17a,18 Dec 6
Mon, November 25 Splitting fields and Galois extensions
Wed, November 27 Thanksgiving Holiday; No Classes
Fri, November 29 Thanksgiving Holiday; No Classes
Mon, December 2 The Fundamental Theorem of Galois Theory
Wed, December 4 Composite extensions, radical extensions.
Fri, December 6 Solvable groups. Solvability by radicals.


You are expected to attend every class and arrive on time. It is your responsibility to be informed of any announcements made in class.