The Number of Irreducible Polynomials over a Finite Field : An Algebraic Proof
Ricky Aditya · International Journal of Applied Mathematics & Statistics/International journal of applied mathematics and statistics · 2015
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 p^n always exists.