Concrete Abstract Algebra: From Numbers to Gröbner Bases
This book introduces basic notions and results from abstract algebra together with their concrete applications to cryptography and factorization of numbers and polynomials. Its first chapter is called “Numbers”; it starts from scratch and ends with the RSA method and with algorithms for prime factorization. Chapter 2 (Groups) develops basics of group theory and ends with the Sylow Theorems. Chapter 3 (Rings) presents basic facts on commutative rings, with emphasis on unique factorization. Chapter 4 (Polynomials) is a preparation for the final chapter, which is called “Gröbner bases”. There, the classical Hilbert basis theorem is proved using Gröbner bases, and the Buchberger algorithm for computing Gröbner bases is presented in detail, together with simple applications to solving systems of polynomial equations. Unlike many books on abstract algebra, this one is written in a very lively style, with emphasis on illuminating examples and applications. This makes the book a valuable addition to undergraduate literature on this topical subject.