Graduate Algebra 2

UPenn MATH 6030, Spring 2025.

daniel.krashen+alg2@gmail.com

Lecture 1 (1/15/2025): Introductions, polynomials, PIDs and division

Lecture 2 (1/22/2025): Localization and unique factorization domains

Lecture 3 (1/27/2025): Polynomials over UFDs are UFDs, field extensions

Lecture 4 (1/29/2025): More on algebraic extensions, existence of algebraic closure

Lecture 5 (2/3/2025): Galois correspondences, normal extensions

Lecture 6 (2/5/2025): Separability, Galois = separable and normal

Lecture 7 (2/10/2025): The Galois correspondence, inseparable extensions

Lecture 8 (2/12/2025): Transcendental extensions, linear disjointness

Highlights: Isaacs Thm 19.4, Jacobson (BAII) II.8.7, Linear disjointness from Jacobson (BAII) II.8.15, simple transcendental extensions are totally transcendental (Isaacs Cor 17.8)

Lecture 9 (2/17/2025): Separable and inseparable closures, more on linear disjointness

For another reference on tensor products, see Jacobson's Basic Algebra II, 3.7

Lecture 10 (2/19/2025): Galois descent

Lecture 11 (2/24/2025): Twisted forms

Lecture 12 (2/26/2025): Classifying separable and Galois algebras, infinite Galois theory

Lecture 13 (3/3/2025): Summary statement of twisted forms, review for exam 1

Lecture 14 (3/17/2025): Some ideal theory, introduction to factorization of ideals

Lecture 15 (3/19/2025): Noether-Lasker, associated primes

Lecture 16 (3/24/2025): Overview of Krull's principal ideal theorem

Lecture 17 (3/26/2025): Krull's principal ideal theorem and the generalized Cayley-Hamilton Theorem / Nakayama Lemma

Lecture 18 (3/31/2025): Cayley-Hamilton-Nakayama continued, integral extensions

Lecture 19 (4/1/2025): Local rings, going up, and first steps to transcendence

Lecture 20 (4/6/2025): Matroids, transcendence and Noether normalization

Lecture 21 (4/8/2025): Introduction to Dedekind domains

Lecture 22 (4/14/2025): Dedekind domains, continued

Lecture 23 (4/16/2025): Some comments about Hopf algebras. Dedekind domains, fin. A few words about symmetric and exterior algebras.

Lecture 24 (4/21/2025): Tensor algebras, symmetric algebras, exterior algebras.