I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an irreducible polynomial $p(t)\in F[t]$.
Valuation is clear, for $q(t)\in F[t]$ it's the maximal $k$ s.t. $q(t)=p(t)^kq'(t)$ where $q(t)$ is the remainder polynomial in $F[t]$.
I know that for $c>1$ I get a non-archimedean absolute value by $|\cdot|=c^{-v(\cdot)}$ where $v(\cdot)$ is a valuation.
Now I could just use my valuation from above and fix some basis greater $1$, e.g. $e$, and get an absolute value but I'm not sure whether that as really p-adic-like. I've read that the choice of $p$ as basis for the valuation $v_p(\cdot)$ is sensible (why is this so?) and hence I would guess that in this case a smarter choice of the basis would be sensible as well.
If $F=\mathbb{C}$, then $p(t)$ would be of the form $(t-\alpha)$ for some $\alpha\in\mathbb{C}^*$ so in this case I would choose $|\alpha|$ as basis but this strategy is of course not applicable for other fields.