\( \newcommand{\cross}{\times} \newcommand{\isom}{\cong} \newcommand{\RP}{\mathbb{RP}} \newcommand{\EE}{\mathbb{E}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\Aut}{\operatorname{Aut}} \)

MA3D5 Galois theory
Term I 2018-2019

Schedule

Week Date of Monday Topics Pages in Siksek Example sheet Comments
1 Oct. 1 Introduction and overview. Revision: Rings and fields. Polynomial rings. Homomorphisms, automorphisms, relative automorphisms, fixed rings. Evaluation homomorphisms. Quotients, first isomorphism theorem. Principal ideals, polynomial rings over fields are PIDs. Maximal ideals, fields. 1 - 6 There are no support classes this week. Many questions were asked (partly by the lecturer!) about whether or not rings must have a unit.
2 Oct. 8 Extensions and subfields. Quotients. Fields of fractions. Generated subfields. Adjoining roots. The subfields of \(\QQ(\sqrt{2}, \sqrt{3})\). Simple extensions. Adjoining a root gives the generated subfield (Prop 21). Splitting fields. Fields of rational functions. 7 - 12 One
3 Oct. 15 Existence of splitting fields (uniqueness postponed). Extensions are vector spaces. Degrees of extensions. Algebraic and transcendental numbers. Algebraic extensions. Finite extensions are algebraic. Monic polynomials, minimal polynomials. Review of Gauss's lemma, Eisenstein's criterion. 12 - 17 Two
4 Oct. 22 The structure of simple algebraic extensions (Prop 41, Thm 42). Primitive elements, degrees. The Tower Law. Extended examples. The algebraic numbers form a field. 17 - 22 Questions asked by students on 2018-10-25.
5 Oct. 29 Algebraic closures and algebraic numbers. Conjugates over \(K\) and Galois conjugates. Automorphisms preserve conjugates. Easy upper bound on \(\#\Aut(L/K)\). Finite field interlude - multiple roots. Linear independence of automorphsims. Hard upper bound on \(\#\Aut(L/K)\). Fixed fields and the Galois correspondence. 23 - 28 Three Questions asked by students on 2018-11-02.
6 Nov. 5 Lower bounds on \(\#\Aut(L/K)\) - proving that automorphisms exist. The isomorphism extension problem and its solution. Splitting fields are unique up to isomorphism. (Prime) cyclotomic fields and their automorphisms. Hard lower bounds. 29 - 32
7 Nov. 12 \( H = \Aut(L / L^H) \). Finite fields, field characteristic, prime subfields. Examples. Separable polynomials, elements, and extensions. Examples. The formal derivative. The GCD of a polynomial and its derivative. Characteristic zero and separability. Normality. Examples. Galois extensions. Criteria equivalent to being Galois. 32 - 40 Four
8 Nov. 19 The relative automorphism group of a Galois extension (aka the Galois group). Examples of Galois and non-Galois extensions, and also of computing the Galois group. Review of normal groups. Statement of the fundamental theorem of Galois theory. The proof begun. 40 - 46
9 Nov. 26 An extended example. The proof (of FTofGT) completed. The classification of finite fields and the Frobenius. Radical expressions. Simple radical extensions, radical extensions in general. Subnormal series for a group, soluble groups. Simple groups. Commutators and the commutator subgroup. 46 - 50, 55 - 56 Five A student pointed out that the assigned questions were missing a hypothesis, namely finiteness of the extensions. I've added this. Interesting question - what happens if you don't add the finiteness hypothesis?
10 Dec. 3 Perfect groups, \(A_5\) is perfect, so not soluble. New soluble groups from old. \(S_5\) is not soluble. Criterion to be \(S_5\). A polynomial with Galois group \(S_5\). Prime cyclotomic fields, and "prime" simple radical extensions. Radical Galois extensions have soluble Galois groups. Ruler-and-compass constructions, impossibility proofs. 50 - 61 Questions asked by students before 2018-12-02.