Math 81/111
Rings and Fields

Last updated June 27, 2016 13:25:41 EDT

## Syllabus

The main focus of the course will be Chapters II, IV.1 - IV.4, V, VI.1- VI.7/8/9 of Lang's text Algebra. This is the first offering out of this textbook, 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 Chapter II Mostly a quick review: Rings (examples, properties, homomorphism theorems), modules (vector spaces as k[x]-modules, group rings), polynomial rings in several variables, division algorithm over commutative rings with 1, polynomials versus polynomial maps, prime and maximal ideals,
Week 2 II.2, II.5, IV.1, IV.2 operations on ideals, correspondence theorem, CRT, irreducibles and prime elements, UFDs, PIDs, Noetherian rings, Euclidean domains
Week 3 IV.2, IV.3, IV.4, random tidbits Gauss's lemma and corollaries, Irreducibility tests, Hilbert's Basis Theorem, Cyclotomic polynomials, start finite and algebraic field extensions
Week 4 V.1, V.3 (part),supplementary material finite, and algebraic extensions, splitting fields, composites and distinguished classes of extensions
Week 5 IV.1, V.2, supplementary material tests for separability, irreducibility of cylotomic polynomials, finite fields, extending embeddings, existence and uniqueness of splitting fields, algebraic closures and uniqueness
Week 6 V.2, V.3, V.4, supplementary compass and straightedge constructions, embeddings, normality for general algebraic extensions, separability, begin Galois theory
Week 7 V.4, VI.1 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 VI.1, VI.2,VI.4, VI.6 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 VI.2, VI.7, class notes Solvability by radicals, Insolvability of the quintic, The general polynomial of degree n

T. R. Shemanske
Last updated June 27, 2016 13:25:41 EDT