Course description: The bulk of the course is about the characterization by Galois of the polynomial equations that are solvable by radicals. The study of this theorem gives the opportunity to learn about fundamental concepts of abstract algebra together with a beautiful application to a problem of classical and elementary flavor. Galois theory arguably has its place in ones general mathematical culture. Familiarity with it is essential in certain areas of number theory, representation theory, algebraic geometry.

Topics covered:

Cubic equations, Cardano's Formula

Field extensions, algebraic elements, degree of a field extension

Automorphisms of field extensions, group actions

Splitting fields, normal extensions

Characteristic of a field, separable extensions

Galois extensions, its various characterizations

Galois correspondence, fundamental theorem of Galois theory, subgroup and subfield lattices

Solvable groups, groups of small order

Galois’ theorem on solvability of equations by radicals, radical extensions, cyclo- tomic extensions, Lagrange resolvents

Symmetric polynomials, the discriminant, quartic polynomials

Finite fields, Frobenius automorphism

Calculating the Galois group, relation to factoring polynomials

Simplicity of finite separable extensions, example of a finite degree field extension with infinitely many intermediate fields