Finite Fields, An Introduction

Introduction

Add 1 to itself again and again in a finite field and the numbers must eventually return to 0, giving a characteristic of p. We're familiar with the integers mod p, written Z_p, and indeed it is a field. But are there larger finite fields based on Z_p?

The answer is yes, and at first this seems counterintuitive. But if you've worked with fields for a long time, it seems almost inevitable. There is no square root of 3 mod 7, so adjoin it, and produce a larger finite field. Then there is no fourth root of 3 mod 7, so adjoin that, making an even larger finite field. Continue this process, building an ascending chain of finite fields.

Finite fields have many applications, including error correcting codes such as Reed Solomon. I hope to include this information on this web site some day.