Kähler differentials and ramification of infinite places Let $L/K$ be an extension of number fields (i.e. finite extensions of $\mathbb{Q}$). The ramification of finite places, i.e. prime ideals, is controlled by the module of Kähler differentials $\Omega_{L/K}$. But what about infinite places? We say that an infinite place $\sigma: K\to \mathbb{R}$ ramifies if it extends to two distinct places $\sigma, \bar \sigma: L\to \mathbb{C}.$ So my question is, can we somehow enhance Kähler differentials so that they tell us something about ramificatioin of those infinite places? Or do we have some other algebraic gadget for that?
 A: At the following link you will find some info on Kahler differentials and rings of integers in number fields:
Why isn't every finite locally free morphism etale?
Note: Any finite extension $K \subseteq L$ is separable, hence $\Omega^1_{L/K}=0$ is zero.
Citation: "For any finite extension  $K \subseteq L$ of number fields, it follows the morphism $\pi: S:=Spec(\mathcal{O}_L) \rightarrow T:=Spec(\mathcal{O}_K)$ is always integral, locally free (and ramified when $K=\mathbb{Q}$). Hence it is seldom etale. There is an open subscheme $U \subseteq T$ with the property that $\pi_U:\pi^{-1}(U) \rightarrow U$ is etale. The fiber  $\pi^{-1}((p))$ in formula F1 above is unramified iff $l_i=1$ for all $i$.
The module of Kahler differentials $\Omega:=\Omega^1_{\mathcal{O}_L/\mathcal{O}_K}$ is zero on the open subscheme $\pi^{-1}(U)$. Note moreover that the field extension
$\mathcal{O}_K/\mathfrak{p} \subseteq \mathcal{O}_L/\mathfrak{q}$
is always separable since it is a finite extension of finite fields (here the ideals $\mathfrak{p}, \mathfrak{q}$ are maximal ideals). The ramified prime ideals in $\mathcal{O}_K$ are related to the discriminant of the extension $L/K$ (see in "Algebraic number theory", page 49, by Neukirch). In Neukirch he defines the discriminant using a basis for $L$ over $K$, but it can also be defined using the module of Kahler differentials $\Omega$."
