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Concept
Square-free polynomial
Summary
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.
In the case of univariate polynomials, the product rule implies that, if p^2 divides f, then p divides the formal derivative f of f. The converse is also true and hence, is square-free if and only if is a greatest common divisor of the polynomial and its derivative.
A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials
where those of the ak that are non-constant are pairwise coprime square-free polynomials (here, two polynomials are said coprime is their greatest common divisor is a constant; in other words that is the coprimality over the field of fractions of the coefficients that is considered). Every non-zero polynomial admits a square-free factorization, which is unique up to the multiplication and division of the factors by non-zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra.
Over a field of characteristic 0, the quotient of by its GCD with its derivative is the product of the in the above square-free decomposition. Over a perfect field of non-zero characteristic p, this quotient is the product of the such that i is not a multiple of p.
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