TABLE OF CONTENTS A square-free polynomial is one without any repeated factors. In this section we examine the square-free concept and describe an algorithm that factors a polynomial that is not square-free in terms of square-free factors. This type of factorization is the easiest one to obtain and is an important first step in the irreducible factorization of polynomials.
The formal definition of a square-free polynomial reflects the origin of the term square-free.
Definition 9.1. Let F be a field. A polynomial u in F [x] is square-free if there is no polynomial v in F [x] with deg (v, x) > 0 such that v 2 u.
Although the definition is expressed in terms of a squared factor, it implies that the polynomial does not have a factor of the form v n with n ? 2 .
The polynomial u=x 2 +3 x+2=( x+1)( x+2) is square-free, while u= x 4+7 x 3+18 x 2+20 x+8=( x+1)( x+2) 3 is not.
The square-free property can also be described in terms of the irreducible factorization of a polynomial.
Theorem 9.3. Suppose that u is a polynomial in F [x] with an irreducible factorization
Then, u is square-free if and only if n i =1 for 1 ?i ? s.
The proof, which is straightforward, is left to the reader (Exercise 4).
Theorem 9.5 (below)...
