# content of polynomial

The content of a polynomial $f$ 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 $\operatorname{cont}(f)$ or $c(f)$.  Coefficient module is a little more general concept.

If $R$ is a unique factorisation domain (http://planetmath.org/UFD) and  $f,\,g\in R[x]$,  the Gauss lemma I implies 11In a UFD, one can use as contents of $f$ and $g$ the http://planetmath.org/node/5800greatest common divisors $a$ and $b$ of the coefficients of these polynomials, when one has  $f(x)=af_{1}(x)$,  $g(x)=bg_{1}(x)$  with $f_{1}(x)$ and $g_{1}(x)$ primitive polynomials.  Then  $f(x)g(x)=abf_{1}(x)g_{1}(x)$,  and since also $f_{1}g_{1}$ is a primitive polynomial, we see that  $c(fg)=ab=c(f)c(g)$. that

 $\displaystyle c(fg)\;=\;c(f)c(g).$ (1)

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

 $\displaystyle c(fg)\;\subseteq\;c(f)c(g)$ (2)

(cf. product of finitely generated 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):

 $\displaystyle[c(f)]^{n}c(fg)\;=\;[c(f)]^{n+1}c(g).$ (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 (http://planetmath.org/ProductOfFinitelyGeneratedIdeals).

## References

• 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