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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Galois theory} \begin{quote}% see also at \emph{[[Galois group]]} \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{ClassicalGaloisTheory}{Classical Galois theory for fields}\dotfill \pageref*{ClassicalGaloisTheory} \linebreak \noindent\hyperlink{GaloisTheoryForFields}{Galois theory of fields}\dotfill \pageref*{GaloisTheoryForFields} \linebreak \noindent\hyperlink{in_terms_of_separable_algebras}{In terms of separable algebras}\dotfill \pageref*{in_terms_of_separable_algebras} \linebreak \noindent\hyperlink{free_modules}{Free modules}\dotfill \pageref*{free_modules} \linebreak \noindent\hyperlink{separable_algebras}{Separable algebras}\dotfill \pageref*{separable_algebras} \linebreak \noindent\hyperlink{separable_closure}{Separable closure}\dotfill \pageref*{separable_closure} \linebreak \noindent\hyperlink{galois_theory_for_separable_algebras}{Galois theory for separable algebras}\dotfill \pageref*{galois_theory_for_separable_algebras} \linebreak \noindent\hyperlink{GaloisTheoryForSchemes}{Galois theory for schemes}\dotfill \pageref*{GaloisTheoryForSchemes} \linebreak \noindent\hyperlink{StatementOfMainTheorem}{Statement of the main theorem}\dotfill \pageref*{StatementOfMainTheorem} \linebreak \noindent\hyperlink{reproducing_classical_galois_theory_of_field_extensions}{Reproducing classical Galois theory of field extensions}\dotfill \pageref*{reproducing_classical_galois_theory_of_field_extensions} \linebreak \noindent\hyperlink{grothendiecks_galois_theory}{Grothendieck's Galois Theory}\dotfill \pageref*{grothendiecks_galois_theory} \linebreak \noindent\hyperlink{GaloisInTopos}{Galois theory in topos theory}\dotfill \pageref*{GaloisInTopos} \linebreak \noindent\hyperlink{reformulation_of_classical_galois_theory}{Reformulation of classical Galois theory}\dotfill \pageref*{reformulation_of_classical_galois_theory} \linebreak \noindent\hyperlink{topostheoretic_galois_theory}{Topos-theoretic Galois theory}\dotfill \pageref*{topostheoretic_galois_theory} \linebreak \noindent\hyperlink{higher_topos_theoretic_galois_theory}{Higher topos theoretic Galois theory}\dotfill \pageref*{higher_topos_theoretic_galois_theory} \linebreak \noindent\hyperlink{la_longue_marche__travers_la_thorie_de_galois}{`La Longue Marche \`a{} travers la Th\'e{}orie de Galois'}\dotfill \pageref*{la_longue_marche__travers_la_thorie_de_galois} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Classical Galois theory classifies [[field extensions]]. It is a special case of a classification of [[locally constant sheaves]] in a [[topos]] by [[permutation representations]] of the [[fundamental groupoid]]/[[fundamental group]]. Even more generally one can define a [[Galois group]] associated to a [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]]. There is an analogue of the Galois correspondence in this setting, see \hyperlink{Mathew14}{Mathew 14}. \hypertarget{ClassicalGaloisTheory}{}\subsection*{{Classical Galois theory for fields}}\label{ClassicalGaloisTheory} We discuss the classical/traditional case of Galois theory, which concerns the classification of [[field extensions]]. Below in \emph{\hyperlink{GaloisTheoryForSchemes}{Galois theory for schemes}} and then in \emph{\href{http://ncatlab.org/nlab/show/Galois+theory#GaloisInTopos}{Galois theory in a topos}} we discuss how this is a special case of a more general concept of Galois theory in a [[topos]]. \hypertarget{GaloisTheoryForFields}{}\subsubsection*{{Galois theory of fields}}\label{GaloisTheoryForFields} \begin{defn} \label{GaloisExtension}\hypertarget{GaloisExtension}{} We call a [[field extension]] $K \subset L$ a \textbf{Galois extension} if $K \subset L$ is algebraic and there exists a subgroup $G \subset Aut(L)$ of the [[automorphism group]] such that $K \simeq L^G$ is the field of elements that are invariant under $G$. \end{defn} \begin{defn} \label{GaloisGroup}\hypertarget{GaloisGroup}{} If $K \subset L$ is a Galois extension, we define the \textbf{[[Galois group]]} to be \begin{displaymath} Gal(L/K) := Aut_K(L) \,. \end{displaymath} \end{defn} This means that we have \begin{displaymath} K \simeq L^{Gal(L/K)} \,. \end{displaymath} \begin{defn} \label{separable}\hypertarget{separable}{} Let $\bar K$ be a fixed [[algebraic closure]] of $K$. If $F \subset K[X] - \{0\}$ is any collection of non-zero [[polynomial]]s, the [[splitting field]] of $F$ over $K$ is the [[subfield]] of $\bar K$ generated by $K$ and the zeros of the polynomials in $F$. We call $f \in K[X]- \{0\}$ \textbf{separable} if it has no multiple zero in $\bar K$. We call $\alpha \in \bar K$ \textbf{separable over $K$} if the [[irreducible polynomial]] $f^\alpha_K$ of $\alpha$ over $K$ is separable. A subextension $L \subset \bar K$ is called \textbf{separable over $K$} if each $\alpha \in L$ is separable over $K$. \end{defn} \begin{defn} \label{normal}\hypertarget{normal}{} We call $L$ \textbf{normal} over $K$ if for each $\alpha \in L$ the polynomial $f^\alpha_K$ splits completely in linear factors in $L[X]$. \end{defn} \begin{theorem} \label{}\hypertarget{}{} Let $K$ be a [[field]] and $L$ a [[subfield]] of $\bar K$ \begin{displaymath} K \subset L \subset \bar K \,. \end{displaymath} Denote by $I$ the set of [[subfields]] $E$ of $L$ for which $E$ is a finite Galois extension of $K$. Then $I$, when [[poset|partially ordered]] by inclusion is a directed poset. The following assertions are equivalent: \begin{enumerate}% \item $L$ is a \hyperlink{GaloisExtension}{Galois extension} of $K$. \item $L$ is \hyperlink{normal}{normal} and \hyperlink{separable}{separable} over $K$. \item There is a set $F \subset K[X] - \{0\}$ of separable polynomials such that $L$ is the \hyperlink{separable}{splitting field} of $F$ over $K$. \item $\coprod_{E \in I} E \simeq L$ \end{enumerate} If these conditions are satisfied, then there is a group [[isomorphism]] \begin{displaymath} Gal(L/K) \simeq {\lim_{\leftarrow}}_{E \in I} Gal(E/K) \,, \end{displaymath} where on the right we have the [[limit]] over the [[poset]] of [[subfield]] of the [[contravariant functor]] $E \mapsto Gal(E/K)$. \end{theorem} Since each group $Gal(E/K)$ is finite, the above isomorphism can be used to equip $Gal(L/K)$ with a [[profinite space|profinite topology]] (i.e. take the limit in the category of [[topological groups]], where each $Gal(E/K)$ has the discrete topology), making it into a [[profinite group]]. We henceforth consider $Gal(L/K)$ as a profinite group in this way. \begin{theorem} \label{}\hypertarget{}{} ([[main theorem of classical Galois theory]]) Let $K \subset L$ be a \hyperlink{GaloisExtension}{Galois extension} of fields with \hyperlink{GaloisGroup}{Galois group} $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$. More precisely, the maps \begin{displaymath} \{E | E\;is\;a\;subfield\;of\;L\;containing\;K\} \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}} \{H|H\;is\;a\;closed\;subgroup\;of\;G\} \end{displaymath} defined by \begin{displaymath} \phi(E) = Aut_E(L) \end{displaymath} and \begin{displaymath} \psi(H) = L^H \end{displaymath} are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$. If $E$ corresponds to $H$, then we have \begin{enumerate}% \item $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$) $[E:K] \simeq index[G:H]$ if $H$ is open; \item $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as [[topological group]]s); \item for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$; \item $K \subset E$ is Galois precisely if $H$ is a [[normal subgroup]] of $G$; $Gal(E/K) \simeq G/H$ (as [[topological group]]s) if $K \subset E$ is Galois. \end{enumerate} \end{theorem} This appears for instance as \hyperlink{Lenstra}{Lenstra, theorem 2.3}. This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets. This naturally raises the question of what corresponds to non-transitive $G$-sets. \hypertarget{in_terms_of_separable_algebras}{}\subsubsection*{{In terms of separable algebras}}\label{in_terms_of_separable_algebras} \hypertarget{free_modules}{}\paragraph*{{Free modules}}\label{free_modules} Let $A$ be a commutative [[ring]] and $N$ a [[module]] over $A$. A collection of elements $(w_i)_{i \in I}$ of $N$ is called a [[basis]] of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$. If $N$ has a basis it is called \emph{[[free module|free]]} (over $A$). If $N$ is free with basis a [[finite set]] of [[cardinality]] $n$, then we say that $N$ is \emph{free with [[rank]] $n$} (over $A$). In this case, $N$ is a [[finitely generated]] free module. Let $N$ be a finitely generated free $A$-module with basis $w_1, w_2, \cdots, w_n$ and let $f\colon N \to N$ be $A$-linear. Then \begin{displaymath} f(w_i) = \sum_{j = 1}^n a_{i j} w_j \;\;\; (1 \leq i \leq n) \end{displaymath} for certain $a_{i j} \in A$, and the [[trace]] $Tr(f)$ of $f$ is defined by \begin{displaymath} Tr(f) = \sum_{i = 1}^n a_{i i} \,. \end{displaymath} This is an element of $A$ that only depends on $f$, and not on the choice of basis. It is easily checked that the map $Tr : Hom_A(N,N) \to A$ is $A$-linear. \hypertarget{separable_algebras}{}\paragraph*{{Separable algebras}}\label{separable_algebras} Let $A$ be a [[ring]], $B$ an $A$-[[algebra]], and suppose that $B$ is free with finite rank $n$ as an $A$-module. For every $b \in B$ the map $mult_b\colon B \to B$ defined by $mult_b\colon x \mapsto b x$ is $A$-linear, and the [[trace]] $Tr(b)$ or $Tr_{B/A}(b)$ is defined to be $Tr(mult_b)$. The map $Tr\colon B \to A$ is easily seen to be $A$-linear and to satisfy $Tr(a) = n a$ for $a \in A$. The $A$-[[module]] $Hom_A(B,A)$ (underlying which is the [[hom-set]] in the [[category]] of [[module]]s) is clearly free over $A$ with the same rank as $B$. Define the $A$-linear map $\phi\colon B \to Hom_A(B,A)$ by \begin{displaymath} \phi(x) : y \mapsto Tr(x y) \,, \end{displaymath} for $x, y \in B$. \begin{defn} \label{SeparableAlgebra}\hypertarget{SeparableAlgebra}{} If for an $A$-algebra $B$ the the morphism $\phi$ is an [[isomorphism]] we call $B$ \textbf{separable over $A$}, or a \textbf{free separable $A$-algebra} if we wish to stress the condition that $B$ is finitely generated and free as an $A$-module. \end{defn} \hypertarget{separable_closure}{}\paragraph*{{Separable closure}}\label{separable_closure} Recall the notion of \hyperlink{separable}{separable elements} \begin{defn} \label{SeparableClosure}\hypertarget{SeparableClosure}{} Let $K$ be a [[field]] and $\bar K$ an [[algebraic closure]] of $K$. The [[separable closure]] $K_S$ of $K$ is defined by \begin{displaymath} K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\} \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} We have that $K_S$ is a [[subfield]] of $\bar K$ and that $K_S \simeq \bar K$ precisely if $K$ is a [[perfect field]], in particular if the [[characteristic]] of $K$ is 0. From xyz it follows that the inclusion $K \subset K_S$ is Galois. \end{remark} \begin{defn} \label{AbsoluteGaloisGroup}\hypertarget{AbsoluteGaloisGroup}{} The Galois group $Gal(K_S/K)$ is called the \textbf{[[absolute Galois group]]} of $K$. \end{defn} \hypertarget{galois_theory_for_separable_algebras}{}\paragraph*{{Galois theory for separable algebras}}\label{galois_theory_for_separable_algebras} \begin{theorem} \label{}\hypertarget{}{} Let $K$ be a [[field]] and $\pi_1(Spec K)$ its \hyperlink{AbsoluteGaloisGroup}{absolute Galois group}. Then there is an [[equivalence of categories]] \begin{displaymath} SAlg_K^{op} \simeq \pi_1(Spec K) Set \,. \end{displaymath} \end{theorem} \hypertarget{GaloisTheoryForSchemes}{}\subsection*{{Galois theory for schemes}}\label{GaloisTheoryForSchemes} The classical Galois theory for fields is a special case of a general geometric/[[topos theory|topos theoretic]] statement about [[locally constant sheaves]] and the action of the [[fundamental group]] on their [[fiber]]s. \hypertarget{StatementOfMainTheorem}{}\subsubsection*{{Statement of the main theorem}}\label{StatementOfMainTheorem} \begin{defn} \label{}\hypertarget{}{} A morphism $f : Y \to X$ of [[scheme]]s is a \textbf{finite [[étale morphism]]} if there exists a [[covering]] of $X$ by [[affine scheme|affine]] [[open subset]]s $U_i = Spec A_i$, such that \begin{itemize}% \item for each $i$ the open subschemes $f^{-1}(U_i)$ of $Y$ is affine, \item and equal to $Spec B_i$, where $B_i$ is a free \hyperlink{SeparableAlgebra}{separable} $A_i$-algebra. \end{itemize} In this situation we also say that $f : Y \to X$ is a \textbf{finite \'e{}tale covering} of $X$. A [[morphism]] from a finite \'e{}tale covering $f : Y \to X$ to a finite \'e{}tale covering $g : Z \to X$ is a morphism of schemes $h : Y \to Z$ such that $f = g \circ h$. This defines the [[category]] $FEt_X$ of finite \'e{}tale covers of $X$. \end{defn} \begin{theorem} \label{TheoremGaloisTheoryForSchemes}\hypertarget{TheoremGaloisTheoryForSchemes}{} Let $X$ be a [[connected]] [[scheme]]. Then there exists a [[profinite group]] $\pi_1(X)$ -- the [[fundamental group]] of $X$ -- uniquely determined up to [[isomorphism]], such that the \hyperlink{FiniteEtMorphisms}{category of finite \'e{}tale coverings} $FEex$ is [[equivalence of categories|equivalent]] to the category $Fin \pi_1(X) Set$ of finite [[permutation representation]]s of $\pi_1(X)$ ([[finite set]]s, with the [[discrete topology]], on which $\pi_1(X)$ acts continuously). \end{theorem} This appears for instance as \hyperlink{Lenstra}{Lenstra, main theorem 1.11}. It is fully discussed in [[SGA]]1. The [[profinite group]], $\pi_1(X)$, is often called the \emph{[[étale fundamental group]]} of the connected scheme $X$. In [[SGA1]], Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group $\pi_1(X)$ depends on the choice of a \emph{fibre functor} given by a geometric point of $X$. Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a $\pi_1(X)$-torsor for either version of $\pi_1(X)$. This is important in attacks on [[Grothendieck's section conjecture]]. \begin{example} \label{}\hypertarget{}{} \begin{itemize}% \item The [[disjoint union]] of $n$ copies of $X$ corresponds, under this theorem, to a finite set of $n$ elements on which $\pi_1(X)$ acts trivially. \item The fact that for $X = Spec \mathbb{Z}$ there are no other finite \'e{}tale coverings of $X$ is thus expressed by the group $\pi_1(Spec \mathbb{Z})$ being \emph{trivial} . \item The same is true for $\pi_1(Spec K)$, where $K$ is an [[algebraically closed field]]. \item If $K$ is an arbitrary [[field]], then $\pi_1(Spec K)$ is the [[absolute Galois group]] of $K$; i.e. the [[Galois group]] of the \hyperlink{SeparableClosure}{separable closure} $K_S$ over $K$. In this case theorem \ref{TheoremGaloisTheoryForSchemes} is a reformulation of \hyperlink{ClassicalGaloisTheory}{classical Galois theory}. \item In particular, if $K$ is a finite field, then $\pi_1(Spec K) \simeq \hat \mathbb{Z}$. \item Let $X = Spec A$, where $A$ is the [[ring of integers]] in an [[algebraic number field]] $K$. Let $N$ be the maximal [[algebraic extension]] of $N$ that is [[unramified]] at all non-zero prime ideals of $A$. Then $\pi_1(X)$ is the Galois group of $N$ over $K$. \item More generally, if $a \in A$, $a \neq 0$, then $\pi_1(Spec A[\frac{1}{a}])$ is the Galois group, over $K$, of the maximal algebraic extension of $K$ that is unramified at all non-zero prime ideals of $A$ not dividing $a$. \end{itemize} \end{example} \hypertarget{reproducing_classical_galois_theory_of_field_extensions}{}\subsubsection*{{Reproducing classical Galois theory of field extensions}}\label{reproducing_classical_galois_theory_of_field_extensions} In this section we explain the connection between the main theorem of Galois theory for schemes, theorem \ref{TheoremGaloisTheoryForSchemes}, and \hyperlink{ClassicalGaloisTheory}{classical Galois theory}. We denote by $k$ a [[field]]. It is our purpose to show that the [[opposite category]] of the category of free separable $K$-algebras is equivalent to the category of finite $\pi_1(X)$-sets, for a certain profinite group $\pi_1(X)$. This is a special case of the main theorem \ref{TheoremGaloisTheoryForSchemes}, with $X = Spec K$. In the general proof we shall use the contents of this section only for algebraically closed $K$. In that case, which is much simpler, the group $\pi_1(X)$ is trivial, so that the category of finite $\pi_1(X)$-sets is just the category of finite sets. \begin{theorem} \label{}\hypertarget{}{} (\ldots{}) \end{theorem} \hypertarget{grothendiecks_galois_theory}{}\subsection*{{Grothendieck's Galois Theory}}\label{grothendiecks_galois_theory} In [[SGA1]], Grothendieck introduced an abstract formulation of the above theory in terms of \emph{Galois categories}. A Galois category is a category, $\mathcal{C}$, satisfying a small number of properties together with a \emph{fibre functor} $F: \mathcal{C}\to FinSet$, preserving those properties. The theory is more fully described in the entry on [[Grothendieck's Galois theory]]. \hypertarget{GaloisInTopos}{}\subsection*{{Galois theory in topos theory}}\label{GaloisInTopos} One notices that \hyperlink{ClassicalGaloisTheory}{classical Galois theory} has an equivalent reformulation in [[topos theory]]. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory. \hypertarget{reformulation_of_classical_galois_theory}{}\subsubsection*{{Reformulation of classical Galois theory}}\label{reformulation_of_classical_galois_theory} \begin{prop} \label{}\hypertarget{}{} The [[étale morphism]]s $f : Y \to X$ corresponds precisely to the [[locally constant sheaves]] on $X$ with respect to the [[étale topology]], in that it is equivalently a morphism for which there is an [[étale cover]] $\{U_i \to X\}$ such that $f$ is a [[constant sheaf]] on each $U_i$. For $K$ a [[field]] let $Et(K)$ be its [[small site|small]] [[étale site]]. And \begin{displaymath} \mathcal{E} := Sh(Et(K)) \end{displaymath} the [[sheaf topos]] over it. This topos is a \begin{itemize}% \item [[local topos]]; \item [[locally connected topos]]; \item [[connected topos]]. \end{itemize} Then Galois extensions of $K$ correspond precisely to the [[locally constant object]]s in $\mathcal{E}$. The full [[subcategory]] on them is the [[Galois topos]] $Gal(\mathcal{E}) \hookrightarrow \mathcal{E}$. The Galois group is the [[fundamental group of a topos|fundamental group of the topos]]. \end{prop} \hypertarget{topostheoretic_galois_theory}{}\subsubsection*{{Topos-theoretic Galois theory}}\label{topostheoretic_galois_theory} Accordingly in [[topos theory]] Galois theory is generally about the classification of [[locally constant sheaves]]. The Galois group corresponds to the [[fundamental group of a topos|fundamental group of the topos]] . (\ldots{}) \hypertarget{higher_topos_theoretic_galois_theory}{}\subsubsection*{{Higher topos theoretic Galois theory}}\label{higher_topos_theoretic_galois_theory} In the context of [[higher topos theory]], there are accordingly higher analogs of Galois theory. According to [[shape of an (∞,1)-topos|shape theory]], any [[(∞,1)-topos]] $\mathbf{H}$ has an associated fundamental ∞-groupoid $\Pi(\mathbf{H})$, which in general is a [[pro-space]] whose $1$-truncation $\Pi_1(\mathbf{H})$ is the ordinary fundamental groupoid of the underlying [[1-topos]]. Classical topos-theoretic Galois theory states that locally constant sheaves (of sets) on a locally connected topos are equivalent to representations of $\Pi_1(\mathbf{H})$, i.e., functors $\Pi_1(\mathbf{H})\to Set$. This generalizes to higher topoi as follows: \begin{theorem} \label{}\hypertarget{}{} Let $\mathbf{H}$ be a [[locally n-connected (n+1,1)-topos]], $-1\leq n\leq \infty$. Then there is an equivalence of categories \begin{displaymath} \mathbf{H}^lc \simeq \Fun(\Pi_{n+1}(\mathbf{H}), n Grpd), \end{displaymath} where $\mathbf{H}^{lc}\subset\mathbf{H}$ is the subcategory of locally constant objects. \end{theorem} This generalization of Galois theory is discussed in (\hyperlink{Grothendieck75}{Grothendieck 75}, \hyperlink{Hoyois13}{Hoyois 13}, \hyperlink{Hoyois15}{Hoyois 15}). For further discussion in the case $n=\infty$, see \begin{itemize}% \item [[cohesive (infinity,1)-topos -- structures]] -- \href{http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#GaloisTheory}{Galois theory} \end{itemize} \hypertarget{la_longue_marche__travers_la_thorie_de_galois}{}\subsection*{{`La Longue Marche \`a{} travers la Th\'e{}orie de Galois'}}\label{la_longue_marche__travers_la_thorie_de_galois} Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the \hyperlink{AbsoluteGaloisGroup}{absolute Galois group}, $Gal(\overline{\mathbb{Q}},\mathbb{Q})$ of the rational numbers and its \textbf{geometric} action on moduli spaces of Riemann surfaces. This will (one day be) discussed at [[Long March]]. Other entries that relate to this include [[anabelian geometry]], [[children's drawings]] (in other words \emph{Dessins d'enfants}, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the [[Grothendieck-Teichmuller group]]. The [[anabelian geometry|anabelian]] question is: \emph{how much information about the isomorphism class of an algebraic variety, $X$ is contained in the \'e{}tale fundamental group of $X$?} Grothendieck calls varieties which are completely determined by their \textbf{\'e{}tale fundamental group}, \emph{anabelian varieties}. His anabelian dream was to classify the anabelian varieties in all dimensions over all fields. This can be seen to relate to questions of the \'e{}tale homotopy types of varieties. Tim: I have a feeling that this anabelian question should have a form that generalises to higher dimensions. (Not that I can shed much light on progress in dimension one.) Perhaps there is an anabelian version of the homotopy hypothesis or something of that nature. (\ldots{}) \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[Galois group]], [[motivic Galois group]] \item [[Galois connection]] \item [[categorical Galois theory]], [[Grothendieck's Galois theory]] \item [[Galois cover]] \item [[differential Galois theory]] \item [[Galois cohomology]] \item [[Lascar group]] (a Galois group of first order theories) \item [[fundamental theorem of covering spaces]] \item [[anabelian geometry]] \item [[geometric homotopy groups in an (∞,1)-topos]], [[fundamental group of a topos]], [[fundamental ∞-groupoid of an (∞,1)-topos]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Lecture notes on the Galois theory for schemes are in \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} Some of the material above is taken from this. \begin{itemize}% \item [[Ravi Vakil]], [[Kirsten Wickelgren]] \emph{Universal covering spaces and fundamental groups in algebraic geometry as schemes}, (\href{http://math.stanford.edu/~vakil/files/VW2Dec1910.pdf}{pdf}). \end{itemize} A comprehensive textbook is \begin{itemize}% \item [[Francis Borceux]], [[George Janelidze]], \emph{Galois theories} Cambridge University Press (2001) \end{itemize} a review of which can be found at [[Galois Theories]]. The locally simply connected case is discussed for instance in \begin{itemize}% \item Marco Robalo, \emph{Galois Theory towards Dessins d'Enfants} (\href{https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf}{pdf}) \end{itemize} Galois theory in a [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]] is developed in \begin{itemize}% \item [[Akhil Mathew]], The Galois group of a stable homotopy theory, \href{http://arxiv.org/abs/1404.2156}{arXiv}. \end{itemize} Galois theory in topos theory \begin{itemize}% \item [[SGA4]], \emph{Expose VIII, Proposition 2.1.} \item [[Marc Hoyois]], \emph{Higher Galois theory} (\href{http://arxiv.org/abs/1506.07155}{arXiv:1506.07155}) \end{itemize} See also \begin{itemize}% \item \emph{\href{http://www.math.upenn.edu/~galois/}{The Galois Theory Web Page}} \item \href{http://en.wikipedia.org/wiki/Galois_theory}{Wikipedia: Galois theory} \item Tam\'a{}s Szamuely, \emph{Galois groups and fundamental groups}, Cambridge Studies in Adv. Math. \item [[Alexander Grothendieck]], letter to [[Larry Breen]] 12/2/1975 (\href{http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/breen1.html}{web}) \end{itemize} category:Galois theory [[!redirects Galois theories]] [[!redirects Galois extension]] [[!redirects Galois extensions]] \end{document}