Primitive part and content: Difference between revisions

Revision as of 02:43, 12 July 2016

Revision as of 02:50, 12 July 2016

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:<math>P=c(P)\operatorname{pp}(P).</math>

This shows that every polynomial over the rationals is [[associate elements|associated]] towith a unique primitive polynomial over the integers, and that the [[Euclidean algorithm]] allows the computation of this primitive polynomial.

A consequence is that factoring polynomials over the rationalrationals is equivalent to factoring primitive polynomials over the integers. As polynomials with coefficients in a [[field (mathematics)}|field]] are more common than polynomials with integer coefficientcoefficients, it may seem that this equivalence may be used for factoring polynomials with integer coefficients. In fact, thisthe truth is exactly the contraryopposite: every known efficient algorithm for factoring polynomials with rational coefficient uses this equivalence for reducing the problem [[modular arithmetic|modulo]] some prime number {{math|''p''}} (see [[Factorization of polynomials]]).

This equivalence is also used for computing [[polynomial greatest common divisor|greatest common divisors]]s of polynomials, although the [[Euclidean algorithm]] is defined for polynomials with rational coefficients. In fact, in this case, the Euclidean algorithm requires one to compute the [[irreducible fraction|reduced form]] of many fractions, and this makes the Euclidean algorithm less efficient than algorithms which work only with polynomials over the integers (see [[polynomialPolynomial greatest common divisor]]).

==Over a field of fractions==