# 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 $\mathrm{cont}(f)$ 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 ^{1}^{1}In 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 $f(x)=a{f}_{1}(x)$, $g(x)=b{g}_{1}(x)$ with ${f}_{1}(x)$ and ${g}_{1}(x)$ primitive polynomials^{}. Then $f(x)g(x)=ab{f}_{1}(x){g}_{1}(x)$, and since also ${f}_{1}{g}_{1}$ is a primitive polynomial, we see that $c(fg)=ab=c(f)c(g)$. that

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

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

$c(fg)\subseteq c(f)c(g)$ | (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):

${[c(f)]}^{n}c(fg)={[c(f)]}^{n+1}c(g).$ | (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 |