Exercises of Galois Theory E-Book

Simone Malacrida

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

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
polynomials and symmetrical forms
Cauchy modules and monodromy
Binomial, second-, third- and fourth-degree equations.
Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Leseprobe

Table of Contents

"Exercises of Galois Theory"

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

"Exercises of Galois Theory"

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

polynomials and symmetrical forms

Cauchy modules and monodromy

Binomial, second-, third- and fourth-degree equations.

Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Simone Malacrida (1977)

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

ANALYTICAL INDEX

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INTRODUCTION

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I – THEORETICAL OUTLINE

Symmetric polynomials and Cauchy moduli

Galois group

Binomial equations

Solvability by radicals

Fundamental theorem

Solving quadratic equations

Solving third degree equations

Solving quadratic equations

Ruffini-Abel theorem

Monodromy

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II – EXERCISES

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

INTRODUCTION

In this workbook some examples of calculations related to the Galois theory are carried out.

Furthermore, the main theorems used in this theory are presented.

Galois theory is a very powerful formalism for solving any equation of degree less than the fifth both in the real field and in the complex field.

Moreover, some polynomial properties are clarified only by considering this theory which is a particularization of advanced algebra and of group theory.

In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.

What is presented in this workbook is generally covered in advanced algebra courses at the university level.

I

THEORETICAL OUTLINE

Symmetric polynomials and Cauchy moduli

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A polynomial in n variables with coefficients in a field is said to be symmetric in the variables if it happens that:

Where sigma indicates the permutations of the symmetric group S (Galois group, for the definition see the next chapter).

Of all the symmetric polynomials in the single variables, the elementary symmetric functions have particular importance in which each single permutation is given by the sum of the products of the indices.

In this case, the first permutation is simply the sum of the n variables, the second permutation the sum of the two-by-two products, the third the three-by-three products, and so on.

The sign of each permutation is negative for odd powers and positive for even powers.

It can be shown that the elementary symmetric functions are algebraically independent in the field , i.e.:

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