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For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$.

What is the origin of the notations $e$ and $f$?

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That's a good question. While we're at it, I'd also like to know if the formula $\sum_i e_if_i = [K:F]$, which expresses the fact that the "total degree" is the sum of the "local degrees", has a name. I have never seen it called anything but "Formula X.X" or "Theorem X.X". – Bruno Joyal Jul 28 '11 at 7:24
If the prime ideals considered are ideals of a Dedekind domain, then the equality Bruno mentions is a particular case of the so called "fundamental inequality" of valuation theory, where the field degree bounds the sum on the left from above. – Hagen Knaf Jul 28 '11 at 7:50
As for the ramification index: the letter $e$ probably abbreviates the word "exponent", because $e(Q|P)$ is the exponent of $Q$ in the factorization of $P$. The symbol $e$ is used in this sense repeatedly in an article by Dedekind. – Hagen Knaf Jul 28 '11 at 11:03
check hilbert's zahlbericht maybe – yoyo Aug 14 '11 at 1:29

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