The course is an introduction to Galois theory, with the necessary complements of group theory and of the theory of modules over a ring.

Lectures

Written and oral exam

The courses of Linear algebra and Algebra 1.

Modules over a ring. Group actions. Sylow theorems. Soluble groups. Field extensions. Splitting fields. Galois theory.



Modules over a ring. Structure of a finitely generated module over a principal ideal domain. Applications: Jordan canonical form and rational canonical forms.



Group actions. Sylow theorems and applications. Semidirect products.

Soluble groups.



Field extensions. Splitting fields: existence and unicity. Galois correspondence. Normal extensions. Separable and inseparable extensions. Galois extensions. The fundamental theorem of Galois theory.

Primitive Element Theorem. Galois theory for finite fields. Cyclotomic polynomials and their irreducibility. Galois group of a cyclotomic polynomial. Cyclic extensions. Polynomial solvable by radicals. The general polynomial of degree >4. Equations with integer coefficients which are not solvable by radicals. Cubics and quartics.

I.N. Herstein, Algebra, terza edizione, Editori Riuniti, Roma 1993.



D.J.H. Garling, A Course in Galois Theory, Cambridge University Press

C. Procesi, Elementi di Teoria di Galois, Zanichelli



M.F. Atiyah, I.G. MacDonald, Introduzione all'algebra commutativa, Feltrinelli, 1981.



M. Artin, Algebra, Bollati Boringhieri, Torino 1997.



I.N. Stewart, Galois Theory, second edition, CRC Press.