Galois Theory GAL

Instructor: Dr. Mátyás DOMOKOS

Prerequisite: Some experience of both classical and abstract algebra is necessary. The material of the Classical Algebra course is essential for understanding Galois theory. Also, some knowledge about permutation groups is needed.

Text: Ian Stewart, Galois Theory (Second edition), Chapman and Hall and handouts about the classical approach.

Optional topics for the first 2-3 weeks, until Classical Algebra is over(depending on the background of the students)

Ring theory:  factor rings, Euclidean domains, polynomials over fields and over the integers, maximal ideals in rings, quotient fields.

Lattices:   Partially ordered sets, the two definitions, distributivity, modularity, normal subgroup lattices.

Topics

Field extensions: simple extensions, algebraic and transcendental numbers, the degree of an extension, normality and separability, finite fields.

The Galois group: Normal closure, the Galois correspondence, soluble and simple groups, ruler and compass constructions, solution of equations by radicals.

Examples: transcendental degree, the general polynomial equation, the general Galois group, calculating the Galois group, the regular \$n\$-gon, quadratic and cyclotomic fields.

Ordered fields: the fundamental theorem of algebra.

Optional topics (depending on the background of the students):

Coding theory: Linear codes, cyclic codes, BCH-codes. BCH-codes.