I have some questions about the following statement:
Let $P \subset \mathbb{F}_q(t)$ be a prime, with $q=p^e$ for a prime number $p$ and $f$ a polynomial of degree $n$ with coefficients inside $\mathbb{F}_q(t)$. Let $K$ be the residue field of $P$ and $\overline{f}$ the image of $f$ over $K$.
If $\overline{f}$ is squarefree over $K$, $P$ is unramified over $\mathbb{F}_q(t)[X] / f$.
Until now I have seen ramified/unramified primes only in the case of number fields, so perhaps I am missing something. I would appreciate it if someone could check my thoughts about this statement.
- $P \in \mathbb{F}_q[t]$
- $K = \mathbb{F}_q[t] / (p) = \mathbb{F}_{q^k}[t]$ for some $k$.
Also I would appreciate a definition of ramification/unramified in this context. Now I can not see, why this statement should be true.