TABLE OF CONTENTS If F is a finite field and there exists a positive integer m such that m ? = 0 for every ? ? F, then the least such positive integer m is called the characteristic of F and F is said to have characteristic m.
Let F be a finite field. Then the characteristic of F is a prime.
Proof. Let F = q. F contains the identity element 1. Since F is finite, the elements 1, 1 + 1 = 2, 1 + 1 + 1 = 3, cannot be all distinct. Therefore, there is the smallest number p such that p = 1 + 1 + + 1 ( p times) = 0. This p must be a prime number (for if rs = 0 then r = 0 or s = 0).
Let F be a field. A subset K of F that is itself a field under the operations of F will be called a subfield of F. F is called an extension (field) of K. If K ? F, we say that K is a proper subfield of F. So, GF( p n) has characteristic p and contains GF( p) as a subfield.
