This is the second part of the Modern Algebra sequence. The main topics are rings, especially polynomial rings, and Galois theory.

Provisional syllabus: Each of the topics listed below will occupy roughly one-two weeks of course time.

  1. Rings and ideals, basic notions

  2. Polynomial rings

  3. Modules (basic notions)

  4. Fields of fractions

  5. Principal ideal domains, polynomials over a field

  6. Irreducible polynomials and factorization, Eisenstein polynomials

  7. Field extensions and splitting fields

  8. Galois groups and the main theorems of Galois theory

  9. Applications: finite fields, cyclotomic fields

  10. Applications: solution by radicals, ruler and compass constructions

If time permits, we will cover noetherian rings and modules over a PID.

Prerequisites: Modern Algebra I.

Textbook: Joseph Rotman, Galois Theory.

The book Abstract Algebra by Dummit and Foote (on reserve in the math library) can be used as a reference.

Online resources:

Abstract Algebra: Theory and Applications, by Thomas W. Judson

Notes on Modern Algebra II by Patrick Gallagher

Lecture notes by Robert Friedman

Modern Algebra II review notes by Robert Friedman in three parts: Part 1 Part 2 Part 3

Midterms: October 13, November 12 (in class)

Final: to be announced

Practice exams and notes in various languages (as requested)