Fields, An Introduction

Introduction

This topic assumes some familiarity with rings and vector spaces. You won't need all the theorems in those pages, but you need to know what a ring is, and a ring homomorphism, and an integral domain, and a division ring, and the dimension of a vector space.

A field is a commutative division ring. You can add, subtract, multiply, and divide in a field, and + and * are commutative and associative. Elements have inverses, and multiplication distributes over addition.

The concepts of characteristic, isomorphism, and subfield carry over from rings.

Every field contains the subfield generated by 1. This is either the rationals (characteristic 0), or the integers mod n (characteristic n). Actually we know that n is prime, for if a*b = n, then a and b are zero divisors, elements that are not invertible. Every field is based on Z or Z_p, where p is prime.