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,