Abstract Algebra

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.

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...

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

Autorentext

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.

Inhalt

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

Die/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