TABLE OF CONTENTS Let p>1 be a prime number. In this section we describe an algorithm that finds the irreducible factorization of a polynomial u in Z p [x]. First, we obtain a square-free factorization of u by using a modification of the approach described in Section 9.1, and then complete the process by finding the irreducible factorization of each of the square-free factors. Since the finite field Z p has a richer algebraic structure than the infinite integral domain Z, the irreducible factorization problem is simpler in Z p [x] than in Z [x].
Although the factorization problem in Z p [x] is important in its own right, we introduce it here because it plays an essential role in the factorization algorithm in Z [x].
Notation Convention. In this section, to simplify the notation, we use the ordinary algebraic operators (+ and ) for addition and multiplication of polynomials rather than the more cumbersome notation 
p and 
p introduced for Z p in Section 2.3. For example, for u(x) and v(x) in Z p [x], u(x)+ v(x) refers to the addition of polynomials where the arithmetic of corresponding coefficients of the polynomials is performed in Z p .
We begin by describing a number of algebraic relationships that apply to polynomials in Z p [x].
Theorem 9.20. Let p>1 be a prime number.
