Math 250A, Groups, Rings, and Fields, Fall 2018

Office Hours: by appointment, 815 Evans.

Lectures: Tuesdays and Thursdays 2-3:30pm, 241 Cory.

Class notes (November 15, 2018):

Course number: 22417

Prerequisites: Math 114 or equivalent familiarity with abstract algebra.

Text: Lang, Algebra.

Other resources: Paulin's notes.

Syllabus: Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.

Homeworks: Unless otherwise noted, chapters and exercises are from the course text: Lang, Algebra.

1. Unit 1: Groups (Ch I)
1. Some category theory (I.10-I.12). Exercises: I, #49, 50, 52. Prove the Yoneda Lemma.
2. Groups and group actions (I.1-I.5). Exercises: I, #15, 16, 18, 19, 33, 37, 38, 39, 46, 47, 48, 55.
3. Finite groups, Sylow subgroups (I.4-I.6). Exercises: I, #2, 6, 20, 21, 22, 23, 27, 31, 32, 34, 35, 41.
4. Abelian groups (I.7-1.9). Exercises: I, #1, 7, 24, 26, 36, 42, 43, 44.
2. Unit 2: Representations of finite groups (Ch XVIII)
1. Linear algebra background (XIII.1-XIII.6, XIV.1-XIV.3, XVI.1, XVI.2). Exercises: XIII, #1, 3, 5, 8, 9, 10, 12, 14; XIV, #3, 4, 8, 12, 13, 14.
2. Basics of group representations (XVIII.1-XVIII.7). Exercises: XVIII, #1, 2, 3, 6, 8, 15, 16.
3. GL(2, F_q) (XVIII.12). Exercises: XVIII, #10, 11, 12.

Evaluation: There will be weekly homework, a midterm and a final exam.

Midterm 1, Tuesday, September 18.

Focus: Category theory.

Useful study guides:

• Simmons, An introduction to Category Theory.
• Some notes.
• Some exercises.

Midterm 2, Tuesday, October 30.

Focus: Finite groups and their representations.

Final, take-home.

Focus: representations and characters, algebras and modules.

Some practice material (Google will find many alternatives...):

• Bump's notes.
• Baker's notes.
• Some Morita theory notes.
• Some exercises.
• Some more exercises.
• A compilation of exercises.