# nLab the fundamental group and Galois theory

Let $o$ be a Dedekind domain, let $K:=\mathrm{Quot}\left(o\right)$ denote its quotient field, let $L/K$ be a finite separable field extension of degree $n$, let $O\supset o$ be the integral closure of $o$ in $L$. Then $O$ is in particular a Dedekind domain

Let for

$o\stackrel{i}{\to }O\to L$o\stackrel{i}{\to} O\to L

$f:=\mathrm{Spec}\left(i\right):\mathrm{Spec}\left(O\right)\to \mathrm{Spec}\left(o\right)$ be the induced map between the ring spectra called ramified covering

Let $p\in \mathrm{Spec}\left(o\right)$ be a maximal prime ideal. Then the ideal $\mathrm{pO}$ in $O$ has a unique product decomposition

$\mathrm{pO}={P}_{1}^{{e}_{1}}\dots {P}_{r}^{{e}_{r}}$pO=P_1^{e_1}\dots P_r^{e_r}

with different ${P}_{i}\in \mathrm{Spec}\left(O\right)$. It is custom to introduce some classifying vocabulary depending on the kind of this decomposition and the degree of the field extension ${f}_{i}:=\left[O/{P}_{i}:o/p\right]$: the ${e}_{i}$ are called ramification indices and the ${f}_{i}$ are called inertia degrees (see e.g. the German wikipedia for more details). These satisfy the fundamental identity ${\Sigma }_{i}{e}_{i}{f}_{i}=n$ and every point $p\in \mathrm{Spec}\left(O\right)$ has $\le n$ preimages. If $p$ has $ preimages then $p$ is called ramified.

Let now $L/K$ be galois then the Galois automorphisms $\sigma \in \mathrm{Gal}\left(L/K\right):={\mathrm{Aut}}_{K}\left(L\right)$ (i.e. the automorphisms of $L$ which restrict to the identity on $K$) induce automorphisms of schemes $\mathrm{Spec}\left(\sigma \right)$ and (since $\sigma$ fixes $o$) the diagram

$\begin{array}{ccc}\mathrm{Spec}\left(O\right)& \stackrel{Spec\left(\sigma \right)}{\to }& Spec\left(O\right)\\ {↓}^{f}& ↙\\ \mathrm{Spec}\left(o\right)\end{array}$\array{ Spec(O)&\stackrel{\Spec(\sigma)}{\to}&\Spec(O) \\ \downarrow^f &\swarrow \\ Spec(o) }

commutes. $\tau :=Spec\left(\sigma \right)$ is an example of a cover automorphism (also called cover transformation or Deck transformation). Since $\mathrm{Spec}:\mathrm{Rings}\to \mathrm{Schemes}$ is an equivalence of categories we have an isomorphism

$\mathrm{Gal}\left(L/K\right)\simeq {\mathrm{Aut}}_{\mathrm{Spec}\left(o\right)}\left(\mathrm{Spec}\left(O\right)\right)$Gal(L/K)\simeq Aut_{Spec(o)}(Spec(O))

where the object to the right we have the group of cover automorphism.

One can show that there is a maximal unramified extension (${e}_{i}=1$ and the extensions $O/{P}_{i}:o/p$ being separable) $\stackrel{˜}{K}$ of $K$ and the scheme $\stackrel{˜}{Y}:=\mathrm{Spec}\left(\stackrel{˜}{o}\right)$ where $\stackrel{˜}{o}$ denotes the integral closure of $o$ in $\stackrel{˜}{K}$. Then $f:\stackrel{˜}{Y}\to Y:=\mathrm{Spec}\left(O\right)$ satisfies the axioms of a universal covering and consequently we define on the side of schemes the fundamental group

${\pi }_{1}\left(Y\right):={\mathrm{Aut}}_{Y}\left(\stackrel{˜}{Y}\right)\simeq \mathrm{Gal}\left(\stackrel{˜}{K}/K\right)$\pi_1(Y):=Aut_Y(\tilde Y)\simeq Gal(\tilde K/ K)

## References

• Jürgen Neukirch, algebraic number theory, I.§13.6
Created on August 25, 2012 23:31:17 by Stephan Alexander Spahn (79.227.155.61)