content of polynomial

The content of a polynomialMathworldPlanetmathPlanetmathPlanetmath f may be defined in any polynomial ring R[x] over a commutative ring R as the ideal of R generated by the coefficients of the polynomial.  It is denoted by cont(f) or c(f).  Coefficient module is a little more general concept.

If R is a unique factorisation domain ( and  f,gR[x],  the Gauss lemma I implies 11In a UFD, one can use as contents of f and g the common divisorsMathworldPlanetmathPlanetmath a and b of the coefficients of these polynomials, when one has  f(x)=af1(x),  g(x)=bg1(x)  with f1(x) and g1(x) primitive polynomialsMathworldPlanetmath.  Then  f(x)g(x)=abf1(x)g1(x),  and since also f1g1 is a primitive polynomial, we see that  c(fg)=ab=c(f)c(g). that

c(fg)=c(f)c(g). (1)

For an arbitrary commutative ring R, there is only the containment

c(fg)c(f)c(g) (2)

(cf. product of finitely generatedMathworldPlanetmathPlanetmathPlanetmath ideals).  The ideal c(fg) is called the Gaussian ideal of the polynomials f and g.  The polynomial f in R[x] is a , if (2) becomes the equality (1) for all polynomials g in the ring R[x].  The ring R is a Gaussian ring, if all polynomials in R[x] are .

It’s quite interessant, that the equation (1) multiplied by the power [c(f)]n, where n is the degree of the other polynomial g, however is true in any commutative ring R, thus replacing the containment (2):

[c(f)]nc(fg)=[c(f)]n+1c(g). (3)

This result is called the Hilfssatz von Dedekind–Mertens, i.e. the Dedekind–Mertens lemma.  A generalised form of it is in the entry product of finitely generated ideals (


  • 1 Alberto Corso & Sarah Glaz: “Gaussian ideals and the Dedekind–Mertens lemma” in Jürgen Herzog & Gaetana Restuccia (eds.): Geometric and combinatorial aspects of commutative algebra.  Marcel Dekker Inc., New York (2001).
Title content of polynomial
Canonical name ContentOfPolynomial
Date of creation 2013-11-19 18:51:57
Last modified on 2013-11-19 18:51:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 11C08
Related topic CoefficientModule
Related topic PruferRing
Related topic GaussianPolynomials
Defines content of polynomial
Defines Gaussian ideal
Defines Gaussian polynomial
Defines Gaussian ring