# GALOIS THEORY AND FIELDS THEORY

Academic Year 2023/2024 - Teacher: CARMELO ANTONIO FINOCCHIARO

## Expected Learning Outcomes

The goal of the course is twofold : first to present the Galois theory which has a very important historical and cultural role for a mathematician . Second, to advance students in the understanding of algebra and its methods ; in particular , students will face profound demonstrations , which come into play all the notions ( apparently different ) studied during the first year of Algebra and through exercises in class , students will learn to use the concepts learned and develop reasoning type abstract.

## Course Structure

The teaching will be done on the blackboard in a traditional way. Exercises are periodically carried out to improve learning of the subject. If the teaching is given in mixed or remote mode, they can be introduced the necessary changes with respect to what was previously stated, in order to comply the planned program and reported in the syllabus. Verification of learning can also be carried out electronically, should the conditions require it.

## Required Prerequisites

Basic course of Algebra and Linear Algebra.

## Detailed Course Content

The course presents the basic theory of field extensions (finite extensions, finitely generated, algebraic, separable, normal) and, subsequently, the Galois theory, in the case of finite extensions. Finally, some of the applications of Galois Theory will be given, such as the fundamental theorem of algebra, constructions with ruler and compass and the solvability / non-solvability of polynomial equations. Part I: field extensions. Fields and characteristic; finite extensions; elements algebraic and transcendental; algebraic extensions; finitely generated extensions; splitting field of a polynomial; algebraic closure of a field; finite fields; separable extensions; symmetric polynomials; normal extensions. Part II: Galois theory. Isomorphisms and automorphisms of fields; isomorphisms extensions; Galois group of an extension; galoissiane extensions; fundamental theorem of Galois theory. Part III: applications. fundamental theorem of algebra; Cyclotomic extensions; constructions with ruler and compass; solvable groups; standard track and discriminating; cyclic extensions; Abel-Ruffini theorem; formulas for solutions of cubic equations. Algebraic independence and transcendence basis.

## Textbook Information

1) S. Gabelli, Teoria delle equazioni e teoria di Galois, Spinger, 2008.

2) Lecture notes written by the teacher.

## Course Planning

SubjectsText References
1Gruppi; campi1, 2
2Teoria dei campi; Teoria di Galois 1, 2