Content (algebra)

As content (german content ) of a polynomial over a ring refers to the number of the quotient field of, through which you must divide the polynomial, so the greatest common divisor of the coefficients of the resulting polynomial is a unit in the ring. The dependence on the ring is essential.

Its application has this term in Gauss. This represents the contents of a product of two polynomials with respect to the content of its factors. This result is theoretically very interesting, because we can prove the fact that polynomial rings are unique factorization in finitely many variables over factorial rings in particular bodies. Practically you can also use the set to obtain restrictions of rational roots of a polynomial with integer coefficients. In particular, the candidates for rational zeros can be reduced to a finite number, this can be useful in the factorization of polynomials.

  • 4.1 Contents
  • 4.2 corollaries
  • 4.3 Idea of ​​proof
  • 4.4 Historically
  • 4.5 Application

Definition

For a factorial ring

Be a polynomial with coefficients in an arbitrary factorial ring. Then, the content is, and is hereinafter referred to, in the literature, in some cases, the English term is used. The content is unique up to a unit. Next is set.

For the quotient field over a factorial ring

It is a factorial ring and the quotient field. The elements of the quotient field you can write with the help of the prime elements as follows.

The exponents occur are uniquely determined and one can define

This makes it easy to determine the order of a polynomial with coefficients from the body.

Next can now define the content of over

It should be a maximum set of pairwise not associated primes. For completeness, we have defined then

As in the case of a quotient field, the content is only up to Associated awareness uniquely determined ( a choice of leads to the multiplication of the contents of a unit ).

The two definitions agree for polynomials match over the ring, the second definition is a generalization of the first real.

If it is clear, from which come the ring of coefficients, we also write simply.

Example

Example 1 ( to first definition):

The content of a polynomial with coefficients in

Or even. To summarize, however, as a polynomial with coefficients, we obtain

Or any other rational number other than zero.

Example 2 (For the second definition):

The content of is a polynomial with coefficients in a field of fractions of

Or even. To summarize, however, as a polynomial with coefficients, we obtain

Or any other rational number other than zero.

Comments

Polynomials whose content is a unit mean primitive. With the primitive Share ( engl. primitive part) is called.

A polynomial with coefficients in the quotient field of a factorial ring is exactly then from the polynomial ring if the content is.

Lemma of Gauss

Content

It is a factorial ring and its quotient field, then for

In particular, the product of two primitive polynomials is primitive again.

Corollaries

As Lemma of Gauss also the following four corollaries are often referred to in this statement:

  • The polynomial ring over a factorial ring is factorial.
  • If a non - constant polynomial is irreducible over a factorial ring ( in a variable ), then it is irreducible quotient even over his body.
  • If a monic polynomial has a zero in the quotient field, then this is already in the ring itself
  • The product of two normalized polynomials with rational coefficients has only integer coefficients, if there are already the coefficients of and integral.

Other corollaries are:

  • If a given polynomial from the ring, so every zero can be represented in the quotient field such as a fraction, the denominator is a divisor of the leading coefficient and the counter is a divisor of the absolute element (see also theorem on rational zeros ).
  • The primes in the polynomial ring over a factorial ring are precisely the prime elements of the ring together with the primitive prime elements of the polynomial ring over the field of fractions of.
  • Is a factorial ring, then the polynomial ring in finitely many variables is factorial

Idea of ​​proof

First, you convince yourself that this is true for. One can assume, then, that primitive (ie ), and thus must only this special case of the theorem show. It can be seen easily that

Then the theorem is trivial but, because and thus is an integral domain, because a prime ideal.

First Corollary:

It proves that all prime elements of the ring and all primitive prime elements of prim are. If taking advantage of the fact that as Euclidean ring is factorial, we can write each element of the product of these primes ( this had to be shown ). The other corollaries do not require proof idea. You just have to prove the statements directly.

Historically

Gauss shows itself in the Disquisitiones Arithmeticae ( art. 42), the variant:

  • The product of two normalized polynomials with rational coefficients has only integer coefficients, if there are already the coefficients of and integral.

Application

  • Is not divisible into, because the contents of 1 and 3
  • Has no rational zeros, because the only possible rational zeros would be Gaussian and.
  • Is irreducible as a polynomial in, because it has degree 3 and no rational zeros ( With Gaussian one has only a finite number check ).
  • Is factored as a polynomial in to. This one takes first the following trivial factorizations before ( make primitive and exclude with maximum potency! ):
Unit (ring theory) Unique factorization domain Serge Lang Emil Artin Emmy Noether Siegfried Bosch Michael Artin
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