Rings and Fields

Abstract

This chapter introduces some basic concepts of elementary abstract algebra: groups, rings and commutative rings, fields, ideals, cosets of ideals and quotient rings. The immediate objective is to describe modular arithmetic as addition and multiplication in the ring $\mathbb {Z}/m\mathbb {Z}$, of cosets of integers modulo the ideal consisting of all multiples of the modulus m. Doing so places modular arithmetic on a firm theoretical foundation. One consequence is to show that if m is a prime number, then doing modular arithmetic modulo p is the same as working in $\mathbb {Z}/p\mathbb {Z}$, and the latter is a field. So concepts and results involving polynomials (in Chap. 6) and matrices (in Chap. 7) make sense when the “numbers” are integers modulo a prime number p.