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Square-free Polynomial
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Any Separable Polynomial is square-free. Conversly, if the field ''F'' is Perfect , all square-free polynomials over ''F'' are separable. In particular, if ''f'' is a square-free polynomial over a perfect field, then the Greatest Common Divisor of ''f'' and its Formal Derivative ''f'' ′ is 1.

A square-free factorization of a polynomial is a factorization into powers of square-free factors, i.e:

:
f(x) = a_1(x) a_2(x)^2 a_3(x)^3 \cdots a_n(x)^n


where the a_k(x) are pairwise Coprime square-free polynomials. Clearly, any non-zero polynomial admits a square-free factorization, since it could be Factored into Irreducible factors and the Multiplicity of each irreducible factor counted to determine which a_k(x) it is part of.

The utility of a square-free factorization is that it is generally easier to compute than a full irreducible factorization. For this reason, square-free factorization is often used as the first step in polynomial factorization or root-finding algorithms.

Over fields of characteristic 0, only differentiation, polynomial division, and GCD calculation (which can be done using the Euclidean Algorithm ) is required to compute the square-free factorization. Let ''f'' be a non-zero polynomial, decomposed into square-free factors as above. Consider any irreducible factor ''q'' of ''f'': we may write f=q^kh, where ''k''>0 and q
mid h. By the chain rule,
:f'=k\,q^{k-1}q'h+q^kh'.
As the characteristic is 0, ''q'' does not divide ''k'', ''q''′, or ''h'', thus q^k
mid\gcd(f,f') and q^{k-1}\mid\gcd(f,f'). That is, the multiplicity of any irreducible factor in \gcd(f,f') is one less than its multiplicity in ''f'', so

:\gcd(f,f') = a_2a_3^2 \cdots a_n^{n-1} and rac{f}{\gcd(f,f')}= a_1a_2\cdots a_n.

Now, if we compute recursively

:f_0=f\,, f_1=\gcd(f_0,f_0')\,, f_2=\gcd(f_1,f_1')\,, f_3=\gcd(f_2,f_2')\,, \dots

we obtain the polynomials

:g_k:= rac{f_{k-1}}{f_k}=a_ka_{k+1}\cdots a_n,

from which we recover the square-free factors as a_k= rac{g_k}{g_{k+1}}.

A modification of this algorithm also works for polynomials over finite fields, or more generally, perfect fields of non-zero characteristic ''p'', if we know an algorithm to compute ''p''-th roots of elements of the field.