Introduction to Galois Theory E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

The following topics are presented in this book:
symmetric polynomials, symmetric functions, symmetric relations and Cauchy modules
Galois group and Galois theory of equations
binomial equations and fundamental theorem
inverse Galois problem and Ruffini-Abel theorem
resolutions of second, third, and fourth degree equations and monodromy

Leseprobe

Table of Contents

"Introduction to Galois Theory"

INTRODUCTION

BASIC CONCEPTS

GALOIS THEORY OF EQUATIONS

APPLICATIONS OF THE GALOIS THEORY

TOPOLOGICAL VIEW OF THE GALOIS THEORY

"Introduction to Galois Theory"

SIMONE MALACRIDA

The following topics are presented in this book:

symmetric polynomials, symmetric functions, symmetric relations and Cauchy modules

Galois group and Galois theory of equations

binomial equations and fundamental theorem

inverse Galois problem and Ruffini-Abel theorem

resolutions of second, third, and fourth degree equations and monodromy

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

––––––––

INTRODUCTION

––––––––

I – BASIC CONCEPTS

Symmetric polynomials and Cauchy moduli

Symmetrical relationships

––––––––

II - GALOIS THEORY OF EQUATIONS

Galois group

Fundamental modules and reduction of the Galois group

Property of the Galois group

Extension for abelian groups and primitive groups

––––––––

III - APPLICATIONS OF THE GALOIS THEORY

Binomial equations

Solvability by radicals

Fundamental theorem

The inverse G alois problem

More results

Solving quadratic equations

Solving third degree equations

Solving quadratic equations

Ruffini-Abel theorem

Monodromy

––––––––

IV - TOPOLOGICAL VIEW OF THE GALOIS THEORY

Introduction and definitions

Results of the theory

INTRODUCTION

Galois theory is based on a different interpretation of the elementary algebraic equations and extends in a very general way results coming from disparate disciplines: from geometric ones such as the construction of regular polynomials with straightedge and compass to topological ones, going to define the very concepts of field and group up to complex analysis.

As such, its elaboration seemed a bit "obscure" in the history of mathematics, a sort of basic incomprehension concealed the ingenious revelations of Galois (which, we recall, by chance, were not appreciated by illustrious contemporary mathematicians of the young talent ).

It took at least another fifty years after the tragic death of Galois to fully understand the revolutionary ideas of the latter and the extraordinary applications of this theory.

It bases its idea on a total abstraction of equations, no longer conceived in algebraic and numerical terms, but as particular elements of topological and algebraic structures not even conceived at the time of Galois.

Precisely for this reason, the exposition of this theory might seem non-intuitive at first glance.

In fact, it is a question of shaking off a vision that has accompanied us throughout the course of our mathematical studies and it is not easy.

But if this leap can be taken, the powerful results of it are self-evident.

A thorough understanding of Galois theory requires concepts of functional analysis, topology, and advanced algebra.

Therefore this manual is aimed at those who have knowledge of these topics at least at the university level.

I