Università Roma Tre

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

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S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).