I'm self-studying finite ramification theory, and I found the following problem:
Let $\Phi$ be a semistandard Ehrhart form (i.e. its entries are weakly increasing). Prove that $\Phi$ has at least one totally ramified subform.
I thought this would be easy at first but quickly ran into trouble. Here's what I tried:
Let $p$ be a prime, and let $S$ be the set of all points in $\mathbb{Z}/p\mathbb{Z}$ that are fixed by the action of $\Phi$. Since $\Phi$ is semistandard, $|S|\geq 1$. Then by the Fixed Point-Subform correspondence theorem, the number of ramified subforms of $\Phi$ equals $|S|$.
However, how can I show that at least one of these subforms is totally ramified (i.e. its image contains no powers of $p$)? The correspondence theorem doesn't immediately yield this. We probably need to again use the fact that $\Phi$ was assumed semistandard. Can anyone provide a hint here on how to proceed?