# Algebra I Fall 2016

The course meets MWF 12:30PM-13:20PM in WEH 7201. If you want to meet, send email to jcumming@andrew.cmu.edu to arrange an appointment.

Homework will be set most Wednesdays, will be due on the following Wednesday, and should be returned graded by the Wednesday after that. Late homework will not be accepted under any circumstances, but the lowest homework score will be dropped. Homework must be typeset (preferably in LaTeX) and submitted by email as a PDF file.

Important note 1: Your homework file should be named according to the scheme "[Your last name]_algebra_[Number of the homework].pdf", so that eg Aaron Aardvark's tenth homework set is "Aardvark_algebra_10.pdf". I won't guarantee that homework not named like this will get graded in a timely fashion.

Important note 2: I generally encourage cooperation on the homework, but you must write up your solutions by yourself and state with whom you have discussed each problem. Cooperation on the tests is cheating and will lead to serious consequences, which can include failing the course and disciplinary action by the university.

## Prerequisites and syllabus

Prerequisites: You should have a firm grasp of undergraduate group theory and undergraduate linear algebra. See the groups handout for specifics on the group theory. From linear algebra we need only the basic facts about linear dependence, dimension, linear transformations and matrices.

Tentative syllabus.

1. Group actions, orbit-stabiliser relation
2. Cauchy's theorem and the Sylow theorems
3. Composition series and the Jordan-Holder theorem
4. Solvable and nilpotent groups
5. Free groups, generators and relations
6. Rings, modules and ideals
7. Chain conditions in rings and modules
8. (Time allowing) Structure theory of noncommutative rings
9. (Time allowing) Group ring and group representations
10. Structure theorem for fg modules over a PID
11. Fields and field extensions
12. Splitting field, normal and separable extensions
13. The Galois group and Galois correspondence
14. Unsolvability of the quintic

## Exams, HW and grading policy

There will be a takehome final. Grades will be assigned according to a formula in which (roughly speaking) homework counts 50 percent and the final counts 50 percent.

## Homework sets and exams

1. HW1 in TeX and PDF. Due by class time on 7 Sep.
2. HW2 in TeX and PDF. Due by class time on 14 Sep.
3. HW3 in TeX and PDF. Due by class time on 28 Sep.
4. HW4 in TeX and PDF. Due by class time on 5 Oct.
5. HW5 in TeX and PDF. Due by class time on 12 Oct.
6. HW6 in TeX and PDF. Due by class time on 19 Oct.
7. HW7 in TeX and PDF. Due by class time on 2 Nov.
8. HW8 in TeX and PDF. Due by class time on 9 Nov.
9. HW9 in TeX and PDF. Due by class time on 30 Nov.
10. FINAL in TeX and PDF.

## Homework set and exam solutions

1. HW1 solutions.
2. HW2 solutions.
3. HW3 solutions.
4. HW4 solutions.
5. HW1 solutions.
6. HW2 solutions.
7. HW3 solutions.
8. HW4 solutions.

## Lecture notes

1. Completing a proof from Wednesday 16.