I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an infinite prime $\mathfrak{B}$ lying over $\mathfrak{p}$, define the inertia degree resp. ramification index by $e_{\mathfrak{B}/\mathfrak{p}}=1$.

But I am confused: in the case where $\mathfrak{p}$ comes from a real embedding $\sigma$, and $\mathfrak{B}$ comes from a complex (non-real) embedding $\tau$ extending $\sigma$, don't we say $\mathfrak{p}$ ramifies in $L$, because both $\tau$ and $\bar{\tau}$ extend $\sigma$?


Add: Neukirch also defines $f_{\mathfrak{B}/\mathfrak{p}}=[L_\mathfrak{B}:K_\mathfrak{p}]$ (loc. cit.) but I think the natural definitions analogues to finite cases should be $f_{\mathfrak{B}/\mathfrak{p}}=\log|\tau(x)|/\log|\sigma(x)|$ for some $x\in K_\mathfrak{p}^\times$, which always equals 1 for $\tau$ extending $\sigma$. It seems Neukirch is using a non-standard convention here? If so is it good for something?

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    $\begingroup$ I think this convention is chosen if you want to have a result like his Proposition 1.2.(i) be true for arbitrary primes, including the infinite primes... Unless you allow e=2 when going from the reals to the complex numbers, but then say f=1 in this case. $\endgroup$ Apr 26, 2014 at 2:47
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    $\begingroup$ Prop 1.2(i) is easily salvaged by saying f=1 and e=2 for complex embeddings extending real ones, but (iii) and (iv) seem to me to require his convention... $\endgroup$ May 28, 2015 at 20:14


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