Gauss’s lemma II

Definition.  A polynomialMathworldPlanetmathPlanetmath P=anxn++a0 over an integral domainMathworldPlanetmath D is said to be primitive if its coefficients are not all divisible by 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 (

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.

Title Gauss’s lemma II
Canonical name GausssLemmaII
Date of creation 2013-03-22 13:07:52
Last modified on 2013-03-22 13:07:52
Owner bshanks (153)
Last modified by bshanks (153)
Numerical id 18
Author bshanks (153)
Entry type Theorem
Classification msc 12E05
Synonym Gauss’ lemma II
Related topic GausssLemmaI
Related topic EisensteinCriterion
Related topic ProofOfEisensteinCriterion
Related topic PrimeFactorsOfXn1
Related topic AlternativeProofThatSqrt2IsIrrational
Defines primitive polynomial