Abstract Algebra

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.

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 text’s brevity, its treatment of groups consists of definitions, examples, and a discussion of subgroups and cosets that culminates in LaGrange’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

• Elementary Number TheoryDivisibility Primes and factorization Congruences Solving congruences Theorems of Fermat and EulerRSA cryptosystem GroupsDe nition of a group Examples of groups Subgroups Cosets and Lagrange's Theorem Rings Defiition of a ring Subrings and idealsRing homomorphisms Integral domains Fields Definition and basic properties of a fieldFinite Fields Number of elements in a finite fieldHow to construct finite fieldsProperties of finite fieldsPolynomials over finite fieldsPermutation polynomials Applications Orthogonal latin squares Die/Hellman key exchange Vector Spaces Definition and examples Basic properties of vector spaces Subspaces PolynomialsBasics Unique factorization Polynomials over the real and complex numbers Root formulas Linear CodesBasics Hamming codes Encoding Decoding Further study Exercises AppendixMathematical induction Well-ordering PrincipleSets Functions Permutations Matrices Complex numbersHints and Partial Solutions to Selected Exercises