View source for Galois group

The automorphism group of a [[Galois extension|Galois extension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431501.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431502.png" />, i.e. the group of all automorphisms of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431503.png" /> leaving the elements of the subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431504.png" /> fixed. The group is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431505.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431506.png" />. The field of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431507.png" /> coincides with the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431508.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g0431509.png" /> is the splitting field of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315010.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315011.png" />, the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315012.png" /> is also called the Galois group of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315013.png" />. These groups are important in the Galois theory of algebraic equations. The computation of the Galois groups for extensions of algebraic number fields is one of the fundamental tasks of algebraic number theory. Finding Galois extensions with an Abelian Galois group (Abelian extensions) is a part of class field theory. Galois groups of algebraic function fields are also a subject of algebraic geometry.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315014.png" /> be a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315015.png" /> be a finite subgroup of the automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315016.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315017.png" /> will then be a Galois extension of the field of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315018.png" />, and the Galois group of this extension is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315019.png" />; moreover, the degree of the extension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315020.png" />, is equal to the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315021.png" />.

The fundamental result on Galois groups is the following theorem, which is sometimes called the main theorem on Galois extensions or the theorem on Galois correspondence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315022.png" /> is a Galois extension of finite degree of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315023.png" />, then there exists a one-to-one correspondence between all subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315024.png" /> of the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315025.png" /> and all subfields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315027.png" /> that contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315028.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315030.png" /> corresponding to each other are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315031.png" /> is the field of invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315033.png" /> is the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043150/g04315034.png" /> (cf. [[Galois correspondence|Galois correspondence]]). This theorem has numerous analogues in many mathematical theories, and can be generalized to extensions of infinite degree (cf. [[Galois topological group|Galois topological group]]). There exists a generalization of the concept of a Galois group to extensions of arbitrary commutative rings, schemes (cf. [[Fundamental group|Fundamental group]]), and also to the case of extensions of skew-fields.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Algebra" , ''Elements of mathematics'' , '''1''' , Springer  (1989)  pp. Chapt. 1–3  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,   "Fundamentals of Galois theory" , Noordhoff  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Jacobson,   "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR></table>



====Comments====
A Galois group can be endowed with the Krull topology, making it a topological group. This topology is discrete if and only if the group is finite. For the Galois correspondence in the case of infinite Galois groups see [[Galois topological group|Galois topological group]].