Gauss's lemma: Difference between revisions

Latest revision as of 19:57, 2 February 2009

Gauss's lemma states that, in 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).

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.

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 ==Statement==
-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:
+Gauss's lemma states that, in a [[fact about::unique factorization domain]], we have the following:
-* A product of [[primitive polynomial]]s is primitive.
+* A product of [[fact about::primitive polynomial]]s is primitive.
-* The [[content of a polynomial|content]] of a product of polynomials is the product of their contents (upto associates).
+* The [[fact about::content of a polynomial|content]] of a product of polynomials is the product of their contents (upto associates).
 ==Related facts==