Introduction to Advanced Algebra E-Book

Simone Malacrida

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

Veröffentlichungsjahr: 2022

Beschreibung

This book covers advanced algebra consisting of:
types of algebra
category theory
groups and group theory
algebraic structures
Galois theory

Leseprobe

Table of Contents

“Introduction to Advanced Algebra”

INTRODUCTION

ADVANCED ALGEBRA

ALGEBRAIC STRUCTURES

GALOIS THEORY

“Introduction to Advanced Algebra”

SIMONE MALACRIDA

This book covers advanced algebra consisting of:

types of algebra

category theory

groups and group theory

algebraic structures

Galois theory

Simone Malacrida (1977)

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

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ANALYTICAL INDEX

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INTRODUCTION

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I – ADVANCED ALGEBRA

Introduction and definitions

Types of algebra

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II – ALGEBRAIC STRUCTURES

Introduction and definitions

Category theory and Euler’s charachteristics

Groups and group theory

Semigroups, groupids, quasi-groups and loops

Monoids, reticles, magmas

Rings

Bodies and fields

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III – GALOIS THEORY

Introduction and definitions

Results of theory della teoria

INTRODUCTION

This short textbook presents topics related to advanced algebra, that is, the discipline that tends to abstract the knowledge of elementary algebra.

For that reason, this book is dedicated only to those who have knowledge at the college level.

Algebra is a powerful mathematical means of abstracting structures that would otherwise be unknown to human thought.

The various types of algebra are a prerequisite to algebraic structures such as groups, fields and rings, which turn out to be the correct setting for many otherwise unknown mathematical problems.

Only in this way has it been possible to find solutions to atavistic questions that have gripped generations of mathematicians for centuries.

The conceptual leap of advanced algebra is by no means intuitive or simple, so much so that one of the founding fathers of that discipline, the young and exuberant Galois, found enormous difficulty in getting his own point of view understood, despite having turned to eminent mathematicians.

It is precisely because of Galois's theory and its later implications that advanced algebra has demonstrated all its enormous power.

I

ADVANCED ALGEBRA

Introduction and definitions

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Advanced algebra includes abstract algebra, that is, the study of algebraic structures such as groups, rings and fields; category theory, which tends to abstract individual algebraic structures; universal algebra, which studies the bases common to all algebraic structures; and the various types of algebras that can be constructed.

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Deferring to the next chapter for the study of algebraic structures and category theory, universal algebra defines an algebra as a set A having a set of operations on A.

An n-ary operation on A is a function that relates n elements of A to a single element of A.

A nullary operation is simply a constant; a unary operation is a function that relates A to A.

A binary operation is said to have an ariety equal to two i.e., it is a function of the Cartesian product AxA that relates back to A.

A binary operation is also called a law of composition, and an algebraic structure having a binary operation is called a magma, the simplest algebraic structure.

Other more complex algebraic structures are defined by two or more binary operations.

Sum and product are examples of two binary operations, while subtraction is not when referring to the set of natural numbers.