Let $\mathbb{Z}_p$ denote the ring of p-adic integers for some prime $p \in \mathbb{Z}$.
What is the absolute value on $\mathbb{Z}_p[X]$ ?
The motivation for the question comes from a proof of Hensel's lemma where we construct a sequence of polynomials $g_n(X) \in \mathbb{Z}_p[X]$ such that $g_n(X) \to g(X)$. I would like to make sure that I understand what exactly $g_n(X) \to g(X)$ means in this context. The proof is found on page 74 in the book "p-adic Numbers" by Gouvea.
Can we just take $\lvert a_o + a_1X + \cdots + a_nX^n\rvert_p:= \underset{0 \leq i \leq n} \max \lvert a_i \rvert_p$ ? This would satisfy the triangle inequality and positive-definiteness. However I am not convinced that the multiplicative property holds.
EDIT: Or perhaps I am mistaken looking for an absolute value in this context. Perhaps I am really looking for a norm, in which case the suggestion should work.
Are you familiar with valuations? The $p$-adic valuation gives a metric on $\mathbb Q$ and this is used a lot. I am unfamiliar with the particular proof, but the ones I have seen have only used valuations on evaluations on polynomials or similar.
In any case, there're possible generalisations of the $p$-adic valuation (socratean question: how would you define it?) to the ring in question, and the most natural one can be made to satisfy the properties you want if I remember correctly.
further edit: As I alluded to and as the topmost comment points out, there is no need to take this too far since the proof doesn't need it.