Gauss lemma

Intheorypolynomials,Gauss lemma relateshighest common factor ofproducttwo polynomialsinteger coefficients tocorresponding hcfs offactors. If welooking at R = P.Q then any common factor ofcoefficientsP will divide allcoefficientsR, by an easy proof. The lemma worksother way round, limitingcommon factorsR. It supplies whatneededconclude thathcf ofcoefficientsRexactlyproduct ofhcfsPfor Q.

A statement thatequivalent: ifhcfPfor Q1, then it1R, also.

Incaseone variable there issimple proofthis. Considerprime number p,tryshow that R mod p (i.e. Rcoefficients reduced tofieldresidues modulo p)not 0. In factdegreeR mod p issumthoseP mod p andQ mod p, whichmore than enough, because weworking infield.

An important consequencethat R can only factorise asproductpolynomialsrational number coefficients, ifalready does into integer polynomials. One sees this by checkingpowers offixed prime p neededclear denominators;same argument works as before,this version can also be calledGauss lemma. It applies torational root theorem.

There isgeneralisationseveral variables.