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**xianghu21** Let $\displaystyle A$ be a commutative ring and let $\displaystyle A[x]$ be the ring of polynomials in an indeterminate $\displaystyle x$, with coeﬃcients in $\displaystyle A$. Let $\displaystyle f = a_0 + a_1x + \cdots + a_nx^n \in A[x]$. $\displaystyle f$ is said to be primitive if $\displaystyle (a_0, a_1, \cdots, a_n)=(1)$. Prove that if $\displaystyle f$, $\displaystyle g \in A[x]$, then $\displaystyle fg$ is primitive iff $\displaystyle f$ and $\displaystyle g$ are primitive.

[This is Atiyah-Macdonald #1.2d]

I am particularly confused with $\displaystyle \Leftarrow$ which I assume uses Gauss Lemma.

For the $\displaystyle \Rightarrow$, I have:

Suppose that $\displaystyle fg$ is primitive. Now suppose that $\displaystyle f$ were not primitive.

Then $\displaystyle (a_0, a_1, \ldots, a_n) = (1)$ so there is a common factor to all of the terms. But then $\displaystyle fg$ would have a common factor as well. Thus $\displaystyle f$ is primitive, and so must $\displaystyle g$ be.