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\begin{document}

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\section*{Galois group}

\begin{quote}%
see at [[Galois theory]] for more


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToEtaleFundamentalGroup}{Relation to \'e{}tale fundamental group}\dotfill \pageref*{RelationToEtaleFundamentalGroup} \linebreak
\noindent\hyperlink{RelationToFrobeniusMaps}{Relation to Frobenius maps}\dotfill \pageref*{RelationToFrobeniusMaps} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[field extension]] one can consider the corresponding automorphism group. The main statement of [[Galois theory]] is that, when the [[field extension]] is Galois, this [[group]] is called the Galois group and its [[subgroup]]s correspond to subextensions of the [[field extension]].

In [[algebraic geometry]], [[Grothendieck]] defined an analogue of the Galois group called the [[etale fundamental group]] of a connected [[scheme]].

Even more generally there is an analogue of the Galois group in [[stable homotopy theory]]. In fact one can define the Galois group of any [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]], and there is an analogue of the [[Galois theory|Galois correspondence]]. In particular one gets a Galois group associated to an [[E-infinity ring spectrum]]. One recovers the Galois group of a [[scheme]] as the Galois group of its [[derived category of quasi-coherent sheaves]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
Let $K\hookrightarrow L$ denote a Galois [[field extension]], then the [[automorphism group]]

\begin{displaymath}
Gal(K\hookrightarrow L):=Aut_K(L)
\end{displaymath}
consisting just of those automorphisms of $L$ whose restriction to $K$ is the identity is called \emph{Galois group of the field extension $K\hookrightarrow L$.}

\end{defn}
Every Galois group $Gal(K\hookrightarrow L)=lim_{K\hookrightarrow E\hookrightarrow L}Gal (K\hookrightarrow E)$ is a [[profinite group|profinite]] [[topological group]] in that it is the limit of the topologically discrete Galois groups of the intermediate finite extensions between $K$ and $L$.

The just defined Galois group is the one occurring in the classical [[Galois theory]] for fields. The analog of the Galois group in [[Galois theory|Galois theory for schemes]] is a [[fundamental group]] (of a scheme) and is rarely called a `'Galois group''.

The Galois group $Gal(K\hookrightarrow K_s)$ of the separable closure of $K$ is called the \emph{[[absolute Galois group]]} of $K$. In this case we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the [[fundamental group]] of the [[scheme]] $Spec K$. In particular the notion of [[fundamental group of a topos|fundamental group (of a point of) a topos]] generalizes that of Galois group. This observation is the starting point and motivating example of [[Grothendieck's Galois theory]] and more generally of that of [[homotopy groups in an (infinity,1)-topos]].

If the scheme, moreover, is a [[group scheme]] (i.e. endowed with a group structure) modules over the Galois group, which are called [[Galois module|Galois modules]], play an important role in [[algebraic number theory]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToEtaleFundamentalGroup}{}\subsubsection*{{Relation to \'e{}tale fundamental group}}\label{RelationToEtaleFundamentalGroup}

For $K$ a [[field]], then the absolute Galois group of $K$ is equivalent to the [[étale fundamental group]]/[[algebraic fundamental group]] of the [[spectrum of a commutative ring|spectrum]] of $K$.

\begin{displaymath}
\pi_1(Spec(K)) \simeq Gal(K_{sep}/K)
  \,.
\end{displaymath}
If $K$ is a [[number field]], write $\mathcal{O}_K$ for its [[ring of integers]], so that $Spec(\mathcal{O}_K)$ is an [[arithmetic curve]]. Then

\begin{displaymath}
\pi_1(Spec(\mathcal{O}_K)) \simeq Gal(K_{alg}^{ur}/K)
  \,,
\end{displaymath}
where $K_{alg}^{ur}$ is the maximal algebraic extension of $K$ that is [[unramified]] at all non-zero [[prime ideals]] (e.g. \hyperlink{Lenstra85}{Lenstra 85, Example 1.12}).

See also at \emph{\href{Galois+theory#StatementOfMainTheorem}{Galois theory -- Statement of the main theorem}}.

\hypertarget{RelationToFrobeniusMaps}{}\subsubsection*{{Relation to Frobenius maps}}\label{RelationToFrobeniusMaps}

For $K$ a [[number field]] then the [[Frobenius maps]] induce canonical elements in the Galois group. See at \emph{\href{Frobenius+morphism#AsElementsOfGaloisGroup}{Frobenius morphism -- As elements of the Galois group}}.

This crucially enters the definition of [[Artin L-functions]] associated with [[Galois representations]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Galois theory]]


\item [[motivic Galois group]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A standard account is

\begin{itemize}%
\item [[Hendrik Lenstra]], \emph{Galois theory for schemes} , Mathematisch Instituut Universiteit van Amsterdam (1985) (\href{http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf}{pdf})

\end{itemize}
For the Galois group in [[stable homotopy theory]], see

\begin{itemize}%
\item [[Akhil Mathew]], The Galois group of a stable homotopy theory, \href{http://arxiv.org/abs/1404.2156}{arXiv}.

\end{itemize}
[[!redirects Galois groups]]

category: Galois theory



\end{document}