Let $K \subseteq L$ be an extension of number fields, let $v$ be an Archimedean place of $K$ and let $w$ be an Archimedean place of $L$ extending $v$. There are in practice three values that one usually assigns to the ramification index $e(w \mid v)$ and to the inertia degree $f(w \mid v)$:
- $e(w \mid v) := 2$ and $f(w \mid v) := 1$ in any case. This is what happens essentially when one excludes the Archimedean primes from a discussion, e.g. with sentences like "Let $S$ be a finite set of places of $K$ containing all the Archimedean ones. Then for all $v \not\in S$...";
- $e(w \mid v) := 2$ and $f(w \mid v) := 1$ if and only if $v$ is real and $w$ is complex, otherwise $e(w \mid v) = f(w \mid v) = 1$. This is a "standard" choice in many textbooks, and comes from the fact that if $K \subseteq L$ is Galois then the decomposition group $D(w \mid v) := \{ \sigma \in \mathrm{Gal}(L/K) \, \colon w \circ \sigma = w \}$ is either trivial or cyclic of order two, and the latter happens if and only if $v$ is real and $w$ is complex (see here);
- $e(w \mid v) := 1$ and $f(w \mid v) := 2$ if and only if $v$ is real and $w$ is complex, otherwise $e(w \mid v) = f(w \mid v) = 1$. This choice is made in the book "Class field theory" by Gras (see page 7), and allows us to talk about complex conjugation as the "Frobenius at infinity". Moreover, it agrees with the usual definition of the ramification index for non-Archimedean places, which is the index of the valuation groups. The motivation is to simplify some of the notation in class field theory, e.g. to have $\mathbb{Q}(\zeta_n)$ as the "standard" ray class fields of $\mathbb{Q}$ (without adding any ramification at $\infty$), and their totally real subfields $\mathbb{Q}(\zeta_n + \zeta_n^{-1})$ as the ray class fields with extra conditions at $\infty$.
I have two questions concerning this:
- Is there any uniform definition of $e$ and $f$ for an extension of valued fields, which would work also for Archimedean absolute values? The most difficult thing is of course to define $f$.
- Do you have other reasons to prefer one of these definitions?