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单词 GausssLemmaII
释义

Gauss’s lemma II


Definition.  A polynomialMathworldPlanetmathPlanetmath P=anxn++a0 over an integral domainMathworldPlanetmathD is said to be primitive if its coefficients are not all divisibleby any element of D other than a unit.

PropositionPlanetmathPlanetmath (Gauss).  Let D be a unique factorization domainMathworldPlanetmath and F its field of fractionsMathworldPlanetmath.  If a polynomial PD[x] is reducible in F[x], then it is reducible in D[x].

Remark. The above statement is often used in its contrapositive form.  For an example of this usage, see this entry (http://planetmath.org/AlternativeProofThatSqrt2IsIrrational).

Proof.  A primitive polynomial in D[x] is by definition divisible by a non invertiblePlanetmathPlanetmath constant polynomial, and therefore reducible in D[x] (unless it is itself constant). There is therefore nothing to prove unless P (which is not constant) is primitive.  By assumptionPlanetmathPlanetmath there exist non-constant S,TF[x] such that  P=ST.  There are elements a,bF such that aS and bT are in D[x] and are primitive (first multiply by a nonzero element of D to chase any denominators, then divide by the gcd of the resulting coefficients in D).  Then aSbT=abP is primitive by Gauss’s lemma I, but P is primitive as well, so ab is a unit of D and  P=(ab)-1(aS)(bT)  is a nontrivial decomposition of P in D[X].  This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Remark.  Another result with the same name is Gauss’ lemma on quadratic residuesMathworldPlanetmath.

From the above proposition and its proof one may infer the

Theorem.  If a primitive polynomial of D[x] is divisible in F[x], then it splits in D[x] into primitive prime factors.  These are uniquely determined up to unit factors of D.

TitleGauss’s lemma II
Canonical nameGausssLemmaII
Date of creation2013-03-22 13:07:52
Last modified on2013-03-22 13:07:52
Ownerbshanks (153)
Last modified bybshanks (153)
Numerical id18
Authorbshanks (153)
Entry typeTheorem
Classificationmsc 12E05
SynonymGauss’ lemma II
Related topicGausssLemmaI
Related topicEisensteinCriterion
Related topicProofOfEisensteinCriterion
Related topicPrimeFactorsOfXn1
Related topicAlternativeProofThatSqrt2IsIrrational
Definesprimitive polynomial
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