Elements of Algebra
Authors
Description
The Franco-Brazilian partnership of Yves Lequain and Arnaldo Garcia resulted in the production of this work, which is a reference for basic algebra courses at universities in Brazil. They designed it based on lecture notes from a course offered annually at the Institute of Pure and Applied Mathematics (IMPA).
The book is structured in three parts and contains exercises. The first serves as a reference for a basic course on ring theory, with some applications to number theory and algebraic geometry. The second focuses on group theory. The last part helps with a course on the theory of finitely generated modules over Euclidean domains, with applications to the theory of linear operators in finite-dimensional vector spaces.
This book is based on the text Algebra: an Introductory Course, published in this same Collection, whose text has undergone extensive revision and has evolved into this new text.
Target audience
Higher education
Name: Elements of Algebra
Author(s): Arnaldo Garcia e Yves Lequain
Pages: 280
Publication: IMPA, 2022
ISBN: 978-65-89124-10-8
Edition: 7
I DIVISION AND FACTORING IN RINGS
Introduction
1 Rings and Domains
1.1 Definitions and Examples
1.2 Rings of Polynomials
1.3 Euclidean Domains
1.4 Homomorphisms of Rings
1.5 Exercises
2 Single Factoring
2.1 Definitions and Examples
2.2 Factoring in Noetherian Domains
2.3 Single Factoring in Rings of Polynomials
2.4 Exercises
3 Polynomials
3.1 Roots and Factors of a Polynomial
3.2 Irreducibility Criteria
3.3 Resultant of two Polynomials
3.4 Symmetric Polynomials
3.5 Hilbert Basis Theorem
3.6 Exercises
4 Applications
4.1 Sums of two squares
4.2 Integer solutions of X2 + Y2 = Z2
4.3 Bezout’s theorem
4.4 Exercises
II GROUPS
5 Basic Group Theory
5.1 Examples of Groups
5.2 Subgroups
5.3 Side Classes and Lagrange’s Theorem
5.4 Normal Subgroups and Quotient Groups
5.5 Homomorphisms of Groups
5.6 Cyclic Groups
5.7 Finite Groups Generated by Two Elements
5.8 Direct Product of Groups
5.9 Semi-Direct Product of Groups
5.10 Permutation Groups
5.11 Exercises
6 Study of a Group via Permutation Representations
6.1 Representation of a Group by Permutations
6.2 Sylow’s Theorem
6.3 Finite p-Groups
6.4 Classification of Simple Groups of Order ≤ 60
6.5 Classification of Groups of Order ≤15
6.6 Properties of A4 and A5
6.7 Exercises
7 Soluble groups
7.1 Jordan-Hölder theorem
7.2 Soluble groups
7.3 Exercises
III MODULES ON EUCLIDEAN DOMAINS
8 Matrices and Finitely Generated Modules
8.1 Diagonalization of Matrices
8.2 Modules of Homomorphisms
8.3 Submodules of a Free Module
8.4 Structure of Finitely Generated Modules
8.5 Exercises
9 Applications
9.1 Structure of Finitely Generated Abelian Groups
9.2 Jordan’s Canonical Form
9.3 Exercises
NOTATION
BIBLIOGRAPHY
TABLE OF CONTENTS