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Difference between revisions of "Galois topological group"

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A Galois group endowed with the Krull topology; the filter base (i.e. a basis of the open neighbourhoods of the identity) of this topology consists of normal subgroups of finite index. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431901.png" /> is a finite Galois extension, the topology of its Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431902.png" /> is discrete. If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431903.png" /> is the union of finite Galois extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431904.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431905.png" />, the (topological) Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431906.png" /> is the projective limit of the finite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431907.png" /> where each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431908.png" /> is given the discrete topology, and is a [[Profinite group|profinite group]], hence a totally-disconnected compact topological group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g0431909.png" /> is the field of invariants for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319010.png" />, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319011.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319012.png" />. The main theorem on finite Galois extensions may be generalized to infinite extensions: There is a one-to-one correspondence between the closed subgroups of the topological Galois group of a Galois extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319013.png" /> and the subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319014.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319015.png" />.
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A Galois group endowed with the Krull topology; the filter base (i.e. a basis of the open neighbourhoods of the identity) of this topology consists of normal subgroups of finite index. If $L/K$ is a finite Galois extension, the topology of its Galois group $G(L/K)$ is discrete. If the field $L$ is the union of finite Galois extensions $K_i$ of a field $K$, the (topological) Galois group $G(L/K)$ is the projective limit of the finite groups $G(K_i/K)$ where each of the $G(K_i/K)$ is given the discrete topology, and is a [[Profinite group|profinite group]], hence a totally-disconnected compact topological group. If $K'$ is the field of invariants for $G(L/K)$, the subgroup $G(L/K')$ is everywhere dense in $G(L/K)$. The main theorem on finite Galois extensions may be generalized to infinite extensions: There is a one-to-one correspondence between the closed subgroups of the topological Galois group of a Galois extension $L/K$ and the subfields of $L$ containing $K$.
  
  
  
 
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Open subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319016.png" /> correspond to subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319017.png" /> that have finite degree over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319019.png" /> is an arbitrary subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319020.png" />, then the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319021.png" /> is Galois (cf. [[Galois extension|Galois extension]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319022.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043190/g04319023.png" />.
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Open subgroups of $G(L/K)$ correspond to subfields of $L$ that have finite degree over $K$. If $H$ is an arbitrary subgroup of $G(L/K)$, then the extension $L/L^H$ is Galois (cf. [[Galois extension|Galois extension]]), and $G(L/L^H)$ is the closure of $H$.

Latest revision as of 13:11, 16 July 2014

A Galois group endowed with the Krull topology; the filter base (i.e. a basis of the open neighbourhoods of the identity) of this topology consists of normal subgroups of finite index. If $L/K$ is a finite Galois extension, the topology of its Galois group $G(L/K)$ is discrete. If the field $L$ is the union of finite Galois extensions $K_i$ of a field $K$, the (topological) Galois group $G(L/K)$ is the projective limit of the finite groups $G(K_i/K)$ where each of the $G(K_i/K)$ is given the discrete topology, and is a profinite group, hence a totally-disconnected compact topological group. If $K'$ is the field of invariants for $G(L/K)$, the subgroup $G(L/K')$ is everywhere dense in $G(L/K)$. The main theorem on finite Galois extensions may be generalized to infinite extensions: There is a one-to-one correspondence between the closed subgroups of the topological Galois group of a Galois extension $L/K$ and the subfields of $L$ containing $K$.


Comments

Open subgroups of $G(L/K)$ correspond to subfields of $L$ that have finite degree over $K$. If $H$ is an arbitrary subgroup of $G(L/K)$, then the extension $L/L^H$ is Galois (cf. Galois extension), and $G(L/L^H)$ is the closure of $H$.

How to Cite This Entry:
Galois topological group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_topological_group&oldid=32456
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article