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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 $f$?

  • 2
    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".2011-07-28
  • 1
    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.2011-07-28
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    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.2011-07-28
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    check hilbert's zahlbericht maybe2011-08-14

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