Let be a commutative ring and let be the ring of polynomials in an indeterminate , with coeﬃcients in . Let . is said to be primitive if . Prove that if , , then is primitive iff and are primitive.
[This is Atiyah-Macdonald #1.2d]
I am particularly confused with which I assume uses Gauss Lemma.
For the , I have:
Suppose that is primitive. Now suppose that were not primitive.
Then so there is a common factor to all of the terms. But then would have a common factor as well. Thus is primitive, and so must be.