Galois Theory GAL

Instructor: Dr. Csaba SZABÓ

Prerequisite: some experience of algebra (rings) is necessary. Also, some knowledge about permutation groups is needed (this can be substituted by taking the AAL course.)

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

Optional topics (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.


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: the general polynomial equation, the general Galois group, calculating the Galois group, the regular n-gon, quadratic and cyclotomic felds.

Ordered fields: the fundamental theorem of algebra.

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

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