MATHEMATICS W4042, Fall 2015
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.
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)
Homework
assignments
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)
Solution
sheets
Notes on the Main Theorem of Galois Theory
Notes (of Z. Dancso and S. Morgan) on Ruler-Compass constructions vs. Origami constructions
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