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 (Second edition), 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.
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