Gauss’s lemma I
There are a few different things that are sometimes called “Gauss’s Lemma”. See also Gauss’s Lemma II.
Gauss’s Lemma I: If is a UFD and and are both primitive polynomials in , so is .
Proof: Suppose not primitive. We will show either or isn’t as well. not primitive means that there exists some non-unit in that divides all the coefficients of . Let be an irreducible factor of , which exists and is a prime element because is a UFD. We consider the quotient ring of by the principal ideal generated by , which is a prime ideal since is a prime element. The canonical projection induces a surjective ring homomorphism , whose kernel consists of all polynomials all of whose coefficients are divisible by ; these polynomials are therefore not primitive.
Since is a prime ideal, is an integral domain, so is also an integral domain. By hypothesis sends the product to , which is therefore the product of and , and one of these two factors in must be zero. But that means that or is in the kernel of , and therefore not primitive.
|Title||Gauss’s lemma I|
|Date of creation||2013-03-22 13:07:49|
|Last modified on||2013-03-22 13:07:49|
|Last modified by||bshanks (153)|