I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample or a proof.
Thank you.
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Sign up to join this communityI'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample or a proof.
Thank you.
If the two polynomials are $f,g$, then the $\mathbb{Z}_{p}$-module $M := \mathbb{Z}_{p}[X]/(f,g)$ is finitely generated (since at least one of $f,g$ is monic) and satisfies $M \otimes_{\mathbb{Z}_{p}} \mathbb{Z}_{p}/(p) = 0$, hence $M = 0$ by Nakayama.