# NUMERICAL SYSTEMS AND GALOIS THEORY

## Learning outcomes of the course unit

To generalize the concept of absolute value and introduce archimedean and not archimedean valuations. To study completions of a field with respect to different absolute values, with particular emphasis on the p-adic fields.

To define Galois groups for separable and normal extensions of fields, to apply the Fundamental theorem of Galois theory (for finite or infinite extensions) to the study of various extensions (radical extensions, constructible, cyclic, abelian, cyclotomic,...).

## Prerequisites

A basic course in Algebra (groups, rings and fields).

During the course we shall present/introduce (if/when necessary) some basic results of group theory (Cauchy's theorem, Sylow's theorem, structure theorem for finitely generated abelian groups,...) and commutative algebra (algebraic extensions, localization, inverse limits,...).

## Course contents summary

The course will deal mainly with the following topics:

1. Absolute values and valuations, completion of a field, p-adic fields Q_p .

2. Algebraic closure of a field, separability and inseparability, normal extensions.

3. Galois theory (finite and infinite extensions), examples and applications.

## Course contents

Absolute values and valuations (archimedean and not archimedean), topologies induced by absolute values and equivalent absolute values, valuations over the rationals (Ostrowski's theorem). Completions, existence and uniqueness of the completion with respect to an absolute value, valuation rings. p-adic fields Q_p , Hensel's Lemma and applications: square roots and roots of unity in Q_p . Structure of the multiplicative group of Q_p , quadratic extensions of Q_p .

Algebraic closure of a field: existence and uniqueness, embeddings of a field in its algebraic closure, extensions of embeddings. Separability and inseparability, separable extensions. Normal extensions, splitting fields.

Galois group of a field extension, Galois groups of a polynomial as a subgroup of the permutation group of its roots, symmetric functions and extensions with Galois group S_n . Fundamental theorem of Galois theory, examples: finite fields, cyclic extensions (Kummer theory and Artin-Schreier extensions), cyclotomic extensions.

Applications: ruler-and-compass constructions, constructible regular polygons (Gauss), radical extensions, polynomials solvable by radicals (Abel's theorem), fundamental theorem of algebra.

Infinite Galois theory: Krull's topology, profinite groups as inverse limits of finite groups, Galois groups of infinite extensions, fundamental theorem of infinite Galois theory.

Inverse problem of Galois theory: constructions of abelian extensions.

During the course we shall present/introduce (if/when necessary) some basic results of group theory (Cauchy's theorem, Sylow's theorem, structure theorem for finitely generated abelian groups,...) and commutative algebra (algebraic extensions, localization, inverse limits,...).

## Recommended readings

F. Q. Gouvea "p-adic numbers" Springer Universitext

J. Neukirch "Algebraic Number Theory" Springer Grund. der Math. Wissen. 322

I. Stewart "Galois Theory" Chapman & Hall/CRC Mathematics

S. Weintraub "Galois Theory" Springer Universitext

I. N. Herstein "Algebra" Editori Riuniti

## Teaching methods

The preferred teaching tool for the development of such knowledge are the lectures. Taking notes is seen as part of the learning process.

## Assessment methods and criteria

The assessment of learning is done in classic way, through the evaluation of an oral interview. In the colloquium, the student must be able to independently conduct demonstrations and solve exercises using an appropriate algebraic language and a proper mathematical formalism.