The Number of Irreducible Polynomials over a Finite Field : An Algebraic Proof

The concepts of finite field are used in coding theory, in which the finite field is the set of alphabets. To construct a finite field of order p(n), where p is a prime integer and n is a natural number, an irreducible polynomial of degree n over Z(p) is needed. In this article, the number of irreducible polynomial of degree n over Z(p) is given and also proved using abstract algebra approach. Because the number is always positive, for any prime integer p and natural number n, an irreducible polynomial of degree n over Z(p) always exists. Moreover, it implies that a finite field of order pn always exists.