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

MA3D5 Galois theory
Term I 2018-2019


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 23 - ?? Three
6 Nov. 5
7 Nov. 12
8 Nov. 19
9 Nov. 26
10 Dec. 3