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Can't we define them as the subgroups of the automorphism group of an arbitrary extension $L/K$ that fixes a prime $Q$ in $\mathcal O_L$ over a prime $P$ in $\mathcal O_K$?

An ideal answer would probably point out some irredeemable flaw that renders the concept trivial/useless. One problem I can see is that $D/I$ might not be surjective onto the galois group of $\mathcal O_L/Q$ but I am not sure if this is fixable or not...

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    $\begingroup$ I guess you could try doing that. WIll anything useful come out though? It might happen for example that $\mathop{Aut}(L/K)$ is trivial, and yet there is non-trivial decomposition of prime ideals. $\endgroup$ – Jyrki Lahtonen May 24 '15 at 20:21
  • $\begingroup$ I guess my question was meant to figure what sort of stuff goes wrong with non normal extensions. An ideal answer would probably be along those lines. I will add this to the question. $\endgroup$ – Asvin May 24 '15 at 20:37

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