Introduction to Modern Algebra II
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.
Abstract Algebra: Theory and Applications, by Thomas W. Judson
Notes on Modern Algebra II by Patrick Gallagher
Midterms: October 13, November 12 (in class)
Final: to be announced
Practice exams and notes in various languages (as requested)
1st week (due September 17)
2nd week (due September 24)
3rd week (due October 1)
4th week (due October 8)
5th week (due October 15)
6th week (due October 22)
7th week (due October 29)
8th week (due November 5)
(Second Midterm: no homework)
9th week (due November 19)
10th week (due December 3)
11th week (due December 10)