Introduction to Algebra
Authors
Description
In order to understand a scientific subject, we need to have some idea about it. Searching for it should be the first target of those who study and research. With this in mind, Adilson Gonçalves set out to organize elementary material of increasing difficulty involving algebra. Based on his classroom experience at the Federal University of Rio de Janeiro (UFRJ), he works on the notions of sets, functions, equivalence relations, rings, bodies, polynomials and groups.
Galois’ Fundamental Theorem (characteristic zero was chosen as the main objective to be achieved). This theorem presents a solution to the problem of determining formulas for expressing the roots of a polynomial using radical expressions. It also requires and applies the notions seen in the book. The classic problems of doubling the cube, squaring the circle and bisecting the angle are dealt with. There is also an enunciation of Gauss’s Theorem, which characterizes the numbers n ≥ 3 whose regular polygons of n-sides can be made with a ruler and compass.
Target audience
Higher education
Name: Introduction to Algebra
Author(s): e Adilson Gonçalves
Pages: 192
Publication: IMPA, 2017
ISBN: 978-85-244-0430-6
Edition: 6
Foreword
Introduction
1 Preliminary notions
1.1 Sets
1.2 Functions
1.3 Equivalence relation
1.4 Cartesian product and binary operation on a set
2 The integers
2.1 Elementary properties
2.2 Good ordering and the division algorithm
2.3 Ideals and M.D.C.
2.4 Prime numbers and maximal ideals
2.5 Unique factorization
2.6 The Z ringsn
3 Rings, ideals and homomorphisms
3.1 Definition and examples
3.2 Sub-rings
3.3 Ideals and quotient rings
3.4 Homomorphism of rings
3.5 The body of fractions of a domain
4 Polynomials in one variable
4.1 Definition and examples
4.2 The division algorithm
4.3 Principal ideals and maximum common divisor
4.4 Irreducible polynomials and maximal ideals
4.5 Single factorization
4.6 Eisenstein’s criterion
5 Algebraic extensions of the rational
5.1 Adjunction of roots
5.2 Decomposition body of a polynomial
5.3 Degree of an extension
5.4 Construction using a ruler and compass
6 Groups
6.1 Definition and examples
6.2 Subgroups and side classes
6.3 Conjugacy classes
6.4 Quotient groups and group homomorphism
6.5 The simplicity of groups An , n ≥ 5
7 Elementary Galois theory
7.1 Galois extensions and normal extensions
7.2 The Galois correspondence
7.3 Solubility through radicals
References
Index