Abstract Algebra Theory and Applications - Computer Science ...

20Finite FieldsFinite fields appear in many applications of algebra, including coding theoryand cryptography. We already know one finite field, Z p , where p is prime. Inthis chapter we will show that a unique finite field of order p n exists for everyprime p, where n is a positive integer. Finite fields are also called Galoisfields in honor of Évariste Galois, who was one of the first mathematiciansto investigate them.20.1 Structure of a Finite FieldRecall that a field F has characteristic p if p is the smallest positive integersuch that for every nonzero element α in F , we have pα = 0. If no suchinteger exists, then F has characteristic 0. From Theorem 14.5 we knowthat p must be prime. Suppose that F is a finite field with n elements.Then nα = 0 for all α in F . Consequently, the characteristic of F mustbe p, where p is a prime dividing n. This discussion is summarized in thefollowing proposition.Proposition 20.1 If F is a finite field, then the characteristic of F is p,where p is prime.Throughout this chapter we will assume that p is a prime number unlessotherwise stated.Proposition 20.2 If F is a finite field of characteristic p, then the orderof F is p n for some n ∈ N.Proof. Let φ : Z → F be the ring homomorphism defined by φ(n) = n · 1.Since the characteristic of F is p, the kernel of φ must be pZ and the image of346