Gauss's lemma: Difference between revisions
(New page: ==Statement== Gauss's lemma states that, in a gcd domain, we have the following: * A product of primitive polynomials is primitive. * The content of a...) |
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==Statement== | ==Statement== | ||
Gauss's lemma states that, in a [[gcd domain]], we have the following: | Gauss's lemma states that, in a [[fact about::gcd domain]] (in particular, in a [[fact about::Bezout domain]] or a [[fact about::unique factorization domain]]), we have the following: | ||
* A product of [[primitive polynomial]]s is primitive. | * A product of [[primitive polynomial]]s is primitive. | ||
Revision as of 17:13, 1 February 2009
Statement
Gauss's lemma states that, in a gcd domain (in particular, in a Bezout domain or a unique factorization domain), we have the following:
- A product of primitive polynomials is primitive.
- The content of a product of polynomials is the product of their contents (upto associates).
Related facts
Applications
- Unique factorization is polynomial-closed: The polynomial ring over a unique factorization domain is again a unique factorization domain. The proof of this is a direct consequence of Gauss's lemma.