I am trying to figure out the valuation of n! with respect to some p-adic norm p. This is part of a proof about the convergence behavior of $x^n /n!$ in the p-adics.
When I try and expand out n into it's value with respect to the p-adic basis i.e. $1, p, p^2 ,p^3....$ I get something pretty nasty to expand of course. Maybe this is the correct way to go about it and it just is something nasty.
Also, if anyone has resources on the p-adics avaialbe online they know, please let me know.
The $p$-adic order of $n!$ is $$ \sum_{j=1}^\infty \left\lfloor \dfrac{n}{p^j} \right\rfloor $$ (a theorem of Legendre).