Abstract Algebra - A Gentle Introduction

Zusatztext As the subtitle implies! those seeking a standard undergraduate text in abstract algebra should look elsewhere. The authors provide readers with a very brief introduction to some of the central structures of algebra: groups! rings! fields! and vector spaces. As an example of the textâ??s brevity! its treatment of groups consists of definitions! examples! and a discussion of subgroups and cosets that culminates in LaGrangeâ??s theorem. There is no mention of group homomorphisms! normal subgroups! or quotient groups. Nonetheless! various applications of the subject not often addressed in traditional texts are treated within this work. It appears that the intent is to provide enough content for readers to comprehend these applications. Just enough elementary number theory is presented to allow a discussion of the RSA cryptosystem. Sufficient material on finite fields is given for a discussion of Latin squares and the Diffie-Hellman public key exchange. Adequate linear algebra topics foster a discussion of Hamming codes. This text will be suitable for an algebra-based course introducing students to abstract mathematical thought or an algebra course with an emphasis on applications.--D. S. Larson! Gonzaga University! Choice magazine 2016 Informationen zum Autor Gary Mullen is Professor of Mathematics, The Pennsylvania State University, where he earned his Ph.D. His main interest is finite fields, and is founder of the journal "Finite Fields and Their Introduction." He is also the Editor of The Handbook of Finite Fields published by CRC Press. James Sellers is Professor and Associate Head for Undergraduate Mathematics, The Pennsylvania State University, where he also earned his Ph.D. He has published many research articles and won awards related to his efforts to advance mathematics education. Klappentext This book introduces the basic notions of abstract algebra to sophomores and perhaps even junior mathematics majors who have a relatively weak background with conceptual courses. It introduces the material with many concrete examples and establishes a firm foundation for introducing more abstract mathematical notions. Zusammenfassung This book introduces the basic notions of abstract algebra to sophomores and perhaps even junior mathematics majors who have a relatively weak background with conceptual courses. It introduces the material with many concrete examples and establishes a firm foundation for introducing more abstract mathematical notions. Inhaltsverzeichnis Elementary Number Theory Divisibility Primes and factorization Congruences Solving congruences Theorems of Fermat and Euler RSA cryptosystem Groups De nition of a group Examples of groups Subgroups Cosets and Lagrange's Theorem Rings Defiition of a ring Subrings and ideals Ring homomorphisms Integral domains Fields Definition and basic properties of a field Finite Fields Number of elements in a finite field How to construct finite fields Properties of finite fields Polynomials over finite fields Permutation polynomials Applications Orthogonal latin squares Di?e/Hellman key exchange Vector Spaces Definition and examples Basic properties of vector spaces Subspaces Polynomials Basics Unique factorization Polynomials over the real and complex numbers Root formulas Linear Codes Basics Hamming codes Encoding Decoding Further study Exercises Appendix Mathematical induction Well-ordering Pri...

List of contents

Elementary Number Theory
Divisibility 
Primes and factorization 
Congruences
Solving congruences 
Theorems of Fermat and Euler
RSA cryptosystem 
Groups
De nition of a group 
Examples of groups
Subgroups
Cosets and Lagrange's Theorem
Rings
Defiition of a ring 
Subrings and ideals
Ring homomorphisms
Integral domains
Fields
Definition and basic properties of a field
Finite Fields
Number of elements in a finite field
How to construct finite fields
Properties of finite fields
Polynomials over finite fields
Permutation polynomials
Applications
Orthogonal latin squares
Di e/Hellman key exchange
Vector Spaces
Definition and examples
Basic properties of vector spaces
Subspaces
Polynomials
Basics
Unique factorization
Polynomials over the real and complex numbers
Root formulas
Linear Codes
Basics
Hamming codes
Encoding
Decoding
Further study
Exercises
Appendix
Mathematical induction
Well-ordering Principle
Sets 
Functions 
Permutations 
Matrices
Complex numbers
Hints and Partial Solutions to Selected Exercises

Report

As the subtitle implies, those seeking a standard undergraduate text in abstract algebra should look elsewhere. The authors provide readers with a very brief introduction to some of the central structures of algebra: groups, rings, fields, and vector spaces. As an example of the textâEUR(TM)s brevity, its treatment of groups consists of definitions, examples, and a discussion of subgroups and cosets that culminates in LaGrangeâEUR(TM)s theorem. There is no mention of group homomorphisms, normal subgroups, or quotient groups. Nonetheless, various applications of the subject not often addressed in traditional texts are treated within this work. It appears that the intent is to provide enough content for readers to comprehend these applications. Just enough elementary number theory is presented to allow a discussion of the RSA cryptosystem. Sufficient material on finite fields is given for a discussion of Latin squares and the Diffie-Hellman public key exchange. Adequate linear algebra topics foster a discussion of Hamming codes. This text will be suitable for an algebra-based course introducing students to abstract mathematical thought or an algebra course with an emphasis on applications.
--D. S. Larson, Gonzaga University, Choice magazine 2016 

Product details

• Authors: Gary L. Mullen, Gary L. (The Pennsylvania State University Mullen, Gary L. Sellers Mullen, Mullen Gary L., James A. Sellers, Sellers James A.
• Publisher: Taylor & Francis Ltd.
• Languages: English
• Product format: Hardback
• Released: 20.12.2016
• EAN: 9781482250060
• ISBN: 978-1-4822-5006-0
• No. of pages: 204
• Series: Textbooks in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, MATHEMATICS / Applied, MATHEMATICS / Algebra / General, Applied mathematics