General Prerequisites: Rings and Modules is essential and Group Theory is recommended.
Course Overview: The course starts with a discussion of the classical problem of solving polynomial equations by radicals. This is followed by the classical theory of Galois field extensions, culminating in some of the classical theorems in the subject: the insolubility of the general quintic equation, the classification of finite fields and the irreducibility of the cyclotomic polynomials over the rational numbers.
Learning Outcomes: Understanding of the relation between symmetries of roots of a polynomial and its solubility in terms of simple algebraic formulae; working knowledge of interesting group actions in a nontrivial context; working knowledge, with applications, of a nontrivial notion of finite group theory (soluble groups); understanding of the relation between algebraic properties of field extensions.
Course Synopsis: Solvability of cubic and quartic equations by radicals. Review of algebraic extensions, the Tower Law, Gauss’ Lemma and Eisenstein’s criterion. Review of groups acting on sets; upper bounds on the size of the Galois group; the theorem of the Primitive Element. Splitting fields and separable extensions. Characterisation of Galois extensions. The fundamental theorem of Galois theory; explicit examples. Solvability by radicals. Normal extensions. Kummer extensions. Techniques for calculating Galois groups: insolubility of certain quintics. Finite fields, the Frobenius automorphism and classification of finite fields. Cyclotomic extensions and the irreducibility of cyclotomic polynomials over the rationals.
