Math 81/111
Abstract Algebra --- Rings and Fields

Last updated January 21, 2017 14:40:00 EST

## Syllabus

The following is a tentative syllabus for the course. This page will be updated irregularly.
On the other hand, the weekly syllabus contained in the Homework Assignments page will always be accurate.

Lectures Sections in Text Brief Description
Week 1 II.1, II.3, IV.1(part) A quick review of rings (examples, properties, homomorphism theorems),
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.2 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 $\mathbb Q$, normality in Galois extensions, composites and liftings of Galois extensions
Week 8 VI.1, VI.2,VI.4, VI.6 Finite fields, irreducibles over $\mathbb F_p$, 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 January 21, 2017 14:40:00 EST