Math 81/111
Rings and Fields

Last updated July 18, 2017 09:28:14 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 a tentative syllabus; 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 $\mathbb Q$, 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 July 18, 2017 09:28:14 EDT