The formula for the solution to a quadratic equation is well-known. There are similar formulae for cubic and quartic equations, but no formula is possible for quintics. The course explains why this happens.
Irreducible polynomials. Field extensions, degrees and the tower law. Extending isomorphisms. Normal field extensions, splitting fields, separable extensions. The theorem of the primitive element. Groups of automorphisms, fixed fields. The fundamental theorem of Galois theory. The solubility of polynomials of degree at most 4. The insolubility of quintic equations. Additional material: Kummer theory, solubility of polynomials with soluble Galois groups. Construction of a radical solution for a soluble polynomial. Tests for solubility of quintics.