Rings and Fields

This accessible introduction to the mathematics of rings and fields shows how algebraic techniques can be used to solve many difficult problems. The book is carefully organized. Prerequisite mathematical skills are introduced at the beginning, and each chapter opens with an introduction to a problem followed by the development of the algebraic techniques necessary for its solution, using concrete mathematical and non-mathematical examples. Although prior knowledge of group theory is not required, a chapter is included which states the axiom for a group and proves the group theoretic results needed in Galois theory. The work is intended for advanced students and researchers in mathematics, computer science, and electronic engineering.

Rings and Fields provides an accessible introduction to rings and fields that will give the reader an appreciation of the power of algebraic techniques to handle diverse and difficult problems. A review of the prerequisite mathematics is given at the start of the book. Dr Ellis presents his ideas clearly and practically. Rather than presenting theory in abstract terms, chapters begin by introducing a problem and then go on to develop the necessary algebraic techniques for its solution in a purposeful, lucid manner, using concrete mathematical and non-mathematical examples. Although prior knowledge of group theory is unnecessary to understand the rest of the book, for those interested there is a chapter which states the axiom for a group and proves the group theoretic results needed in Galois theory.

Preliminaries; Diophantine equations: Euclidean domains; Construction of projective planes: splitting fields and finite fields; Error codes: primitive elements and subfields; Construction of primitive polynomials: cyclotomic polynomials and factorization; Rule and compass constructions: irreducibility and constructibility; Pappus' theorem and Desargues' theorem in projective planes: Wedderburn's theorem; Solution of polynomials by radicals: Galois groups; Introduction to groups; Cryptography: elliptic curves and factorization; Further reading; Index.