Math 81/111
Rings and Fields

Last updated June 17, 2015 12:56:58 EDT

## Syllabus

The main focus of the course will be Chapters 8 and 9 of the text (Rings, Fields, Galois Theory). This is the first offering of this (revamped) course and the textbook is being used for the first time, so expect the syllabus below to evolve over time, the weekly syllabus contained on the Homework Assignments page will always be accurate.

Lectures Sections in Text Brief Description
Week 1 4.4, 4.5, 8.1, 8.2 Mostly a quick review: Rings (examples, properties, homomorphism theorems), modules (vector spaces as k[x]-modules, group rings), polynomial rings in many variables, division algorithm over commutative rings with 1, polynomials versus polynomial maps
Week 2 8.3, 8.4, CRT in 8.6, irreducibles and prime elements, prime and maximal ideals, UFDs, PIDs, Noetherian rings, Euclidean domains
Week 3 8.5, 8.8, 9.1, 9.9(cyclotomic polynomials) Gauss's lemma and corollaries, Irreducibility tests, Hilbert's Basis Theorem, Cyclotomic polynomials, finite and algebraic field extensions
Week 4 9.2, 9.6(part),supplementary material finite, and algebraic extensions, splitting fields. composites and distinguished classes of extensions
Week 5 9.2, 9.3, 9.4, 9.5 tests for separability, irreducibility of cylotomic polynomials, finite fields, extending embeddings, existence and uniqueness of splitting fields, algebraic closures and uniqueness, compass and straightedge constructions
Week 6 9.6, 9.7 embeddings, normality for general algebraic extensions, separability, begin Galois theory
Week 7 class notes Equivalent versions of the FTGT for finite extensions, examples: cyclotomic, biquadratic, x^3 - 2, x^4-2 over Q, normality in Galois extensions, composites and liftings of Galois extensions
Week 8 class notes Finite fields, irreducibles over F_p, finite abelian groups are Galois groups, prime cyclotomic fields and primitive elements, Artin's theorem on characters, norm and trace, solvability by radicals, Hilbert's Theorem 90
Week 9 class notes Solvability by radicals, Insolvability of the quintic, The general polynomial of degree n, Intro to Algebraic Number Theory
Week 10 class notes Intro to Algebraic Number Theory

T. R. Shemanske
Last updated June 17, 2015 12:56:58 EDT