I'm a bit confused concerning tamely ramified covers of arithmetic schemes. I guess they would reduce to tamely ramified extensions of number fields, but they don't seem to do so. Let me elaborate:
First of all, let me recall the standard definition of tamely ramified covers of arithmetic schemes (by which we mean connected, flat, regular schemes of finite type over a Dedekind domain, or $\mathbb{Z}$ if someone prefer this). Let $D$ be a divisor on $S$ of an arithmetic cover $X\to S=\mathrm{Spec}(A)$, where $A$ is a Dedekind domain. Then $X\to S$ is said to be tamely ramified along $D$ if $X\times_S (S\setminus D)\to (S\setminus D)$ is finite and étale and tamely ramified along $D$.
Now, $X$ being connected and étale over $S\setminus D$ means that it is actually a $G:=\mathrm{Aut}_S(X)$-torsor (see Milne, Étale cohomology, page 40), that is, a Galois cover. Since being a $G$-torsor implies that $G$ acts freely and transitively on the fibers, the number of closed points in the fiber is equal to the order of the group $G$. But this means that the fibers all have the same number of points over $S\setminus D$.
Ok, so far so good. Restricting to rings of integers in Galois number fields is where my problem begins. So, let $X=\mathrm{Spec}(\mathfrak{o}_L)\to S=\mathrm{Spec}(\mathfrak{o}_K)$ be a cover of the associated arithmetic schemes to $L/K$.
In this case, a divisor $D$ is a $\mathit{finite}$ sum $\sum_\mathfrak{p} a_\mathfrak{p} \mathfrak{p}$, $a_i\in \mathbb{Z}$. So $X\to S$ being tamely ramified along $D$ means that $L/K$ is at most tamely ramified at the primes in $D$. Outside $D$, the primes can be unramified, inert and completely split (in $L$). $\mathit{However}$, the only possibility, comparing with tamely ramified coverings of arithmetic schemes is that $\mathfrak{p}\in S\setminus D$ is completely split, but there are infinitely many unramified and inert primes by the Cebotarev density theorem.
What the heck am I missing?
