Spahn
the fundamental group and Galois theory (Rev #1, changes)

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Let oo be a Dedekind domain, let K:=Quot(o)K:=Quot(o) denote its quotient field, let L/KL/K be a finite separable field extension, let OoO\supset o be the integral closure of oo in LL. Then OO is in particular a Dedekind domain

Let for

oiOLo\stackrel{i}{\to} O\to L

f:=Spec(i):Spec(O)Spec(o)f:=Spec(i):Spec(O)\to Spec(o) be the induced map between the ring spectra.

Let pSpec(o)p\in Spec(o) be a maximal prime ideal. Then the ideal pOpO in OO has a unique decomposition

pO=P 1 e 1P r e rpO=P_1^{e_1}\dots P_r^{e_r}

with different P iSpec(O)P_i\in Spec(O)

References

  • Jürgen Neukirch, algebraic number theory, I.§13.6

Revision on August 25, 2012 at 21:53:32 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.