Graduate Algebra 2

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

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

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

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

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

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

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

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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)

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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

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

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

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

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

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