Instructor: Dr. Mátyás DOMOKOS

Prerequisites: The material of the Classical Algebra course (complex numbers, divisibility of polynomials), and a little abstract algebra (notion of groups, fields, rings, vector spaces, dimension of a vector space, permutation groups).

Text: Ian Stewart, Galois Theory, Chapman and Hall

Course description: The bulk of the course is about the solution by Galois of the problem of solving polynomial equations by radicals. This gives an outstanding introduction to some intriguing concepts of abstract algebra. So the course may be chosen as an introduction to abstract algebra with the focus on a concrete problem. The main impact in mathematics and science in general of Galois' beautiful discovery is that it became the model for many powerful applications of the notion of symmetry. Familiarity with Galois theory is essential in certain areas of number theory, representation theory, algebraic geometry.

Topics covered:

Cubic equations, Cardano's Formula, discriminant

Field extensions, algebraic elements, degree of a field extension

Splitting fields, simple finite field extensions

Automorphisms of field extensions, normal and separable extensions

Galois correspondence, subgroup and subfield lattices

Solvability of equations by radials, ruler-and-compass constructions

Transcendental extensions, field of fractions, prime subfield

Finite fields, Frobenius automorphism

Ordered fields, algebraically closed fields