content of polynomial
The content of a polynomial may be defined in any polynomial ring over a commutative ring as the ideal of generated by the coefficients of the polynomial. It is denoted by or . Coefficient module is a little more general concept.
If is a unique factorisation domain (http://planetmath.org/UFD) and , the Gauss lemma I implies 11In a UFD, one can use as contents of and the http://planetmath.org/node/5800greatest common divisors and of the coefficients of these polynomials, when one has , with and primitive polynomials. Then , and since also is a primitive polynomial, we see that . that
For an arbitrary commutative ring , there is only the containment
(cf. product of finitely generated ideals). The ideal is called the Gaussian ideal of the polynomials
and . The polynomial in is a , if (2) becomes the equality (1) for all polynomials in the ring . The ring is a Gaussian ring, if all polynomials in
It’s quite interessant, that the equation (1) multiplied by the power , where is the degree of the other polynomial , however is true in any commutative ring , thus replacing the containment (2):
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).
- 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|
|Date of creation||2013-11-19 18:51:57|
|Last modified on||2013-11-19 18:51:57|
|Last modified by||pahio (2872)|
|Defines||content of polynomial|