*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.