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When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of $O_K$. We have the famous relation $n = \sum_{i=1}^r e_i f_i$, where $f_i = [O_K / P_i : \Bbb F_p]$.

Where do these notations $e, f, r$ come from?

Remarks :

  1. The letter $r$ is also denoted for the number $r_1$ (resp $r_2$) of real (resp. pairs of complex) embeddings of $K$. It gives the number $n=r_1 + 2r_2$ (with $2=e(w/v)$ for a complex place $w$, and $f(w/v):=1$) of infinite primes above the unique infinite place of $\Bbb Q$. But why would the number of primes above a given (finite/infinite) prime be denoted by $r$? Maybe the "rank" of some fiber?

  2. It looks like the ramification (Verzweigung) index $e$ denotes "exponent".

  3. The inertia index (inertia degree, Trägheitsgrad) is denoted by the next letter after $e$. Or is it because of "residual field" ?

Thank you!

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    $\begingroup$ Have you learned this material only from books written in relatively recent times? The letter $r$ was not common (I think) in the books I read as a student. Back then it was traditional to use $e_i$, $f_i$, and $g$. The algebraic notion of ramification was developed by analogy to its role in Riemann surfaces (function fields over $\mathbf C$). See if Riemann wrote $e_i$ in his papers, or if Dedekind and Weber did in their joint paper developing function field arithmetic more algebraically (over arbitrary algebraically closed fields). $\endgroup$ – KCd Jul 16 '18 at 20:18
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    $\begingroup$ I also think $r_1$ and $r_2$ are not old-fashioned notations, where they were written as $r$ and $s$ perhaps. Look in Weyl's "Algebraic Theory of Numbers" to see what he wrote. It was one of the earliest books on the subject in English. Also look in Supplement XI of Dirichlet--Dedekind's "Vorlesungen ueber Zahlentheorie". $\endgroup$ – KCd Jul 16 '18 at 20:26
  • $\begingroup$ My first comment should have said at the end "algebraically closed fields of characteristic $0$". $\endgroup$ – KCd Jul 16 '18 at 21:00

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