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