Gauss's lemma - Revision history

Vipul at 19:57, 2 February 2009

2009-02-02T19:57:46Z

← Older revision		Revision as of 19:57, 2 February 2009
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	==Statement==		==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==		==Related facts==

Vipul at 17:13, 1 February 2009

2009-02-01T17:13:11Z

← Older revision		Revision as of 17:13, 1 February 2009
Line 1:		Line 1:
	==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.

Vipul: 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...

2009-02-01T16:46:02Z

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...

New page

==Statement==

Gauss's lemma states that, in a [[gcd domain]], we have the following:

* A product of [[primitive polynomial]]s is primitive.
* The [[content of a polynomial|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.