View source for Schur index

<!--
s0834401.png
$#A+1 = 53 n = 2
$#C+1 = 53 : ~/encyclopedia/old_files/data/S083/S.0803440 Schur index
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The Schur index of a central simple algebra  $  A $
over a field  $  K $(
cf. [[Central simple algebra|Central simple algebra]]) is the degree of the division algebra  $  D $
such that  $  A $
is a full matrix algebra  $  M _ {n} ( D) $
over  $  D $.

Let  $  G $
be a [[Finite group|finite group]],  $  K $
a [[Field|field]] and  $  \overline{K}\; $
the [[Algebraic closure|algebraic closure]] of  $  K $. 
Let  $  V $
be an irreducible  $  \overline{K}\; [ G] $-
module with character  $  \rho $(
cf. [[Irreducible module|Irreducible module]]). Let  $  K( \rho ) $
be obtained from  $  K $
by adjoining the values  $  \rho ( g) $, 
$  g \in G $. 
The Schur index of the module  $  V $, 
$  m _ {K} ( V) $, 
(or the Schur index of the character  $  \rho $) 
is the minimal degree of a field  $  S $
extending  $  K( \rho ) $
such that  $  V $
descends to  $  S $, 
i.e. such that there is an  $  S[ G] $-
module  $  W $
for which  $  V \simeq \overline{K}\; \otimes _ {S} W $.

For a finite field  $  K $
the Schur index is always  $  1 $, 
[[#References|[a1]]].

A basic result on the Schur index is that for each  $  K[ G] $-
module  $  W $
the multiplicity of  $  V $
in  $  \overline{K}\; \otimes _ {K} W $
is a multiple of  $  m _ {K} ( V) $.

A field  $  S \subset  \overline{K}\; $
is a splitting field for a finite group  $  G $
if each irreducible  $  S[ G] $-
module is absolutely irreducible, i.e. if  $  \overline{K}\; \otimes _ {S} V $
is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [[#References|[a1]]] that if  $  d $
is the exponent of a finite group  $  G $(
i.e.  $  d $
is the smallest integer such that  $  g  ^ {d} = e $
for all  $  g \in G $), 
then  $  \mathbf Q ( 1  ^ {1/d} ) $
is a splitting field for  $  G $.

The set  $  S( K) $
of classes of central simple algebras over  $  K $
which occur as components of a group algebra  $  K[ G] $
for some finite group  $  G $
is a subgroup of the [[Brauer group|Brauer group]]  $   \mathop{\rm Br} ( K) $
of  $  K $, 
and is known as the Schur subgroup of  $   \mathop{\rm Br} ( K) $. 
Cf. [[#References|[a4]]] for results on the structure of  $  S( K) $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brauer,   "On the representation of a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344054.png" /> in the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083440/s08344055.png" />-th roots of unity"  ''Amer. J. Math.'' , '''67'''  (1945)  pp. 461–471</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)  pp. §90, §41</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,   "Finite groups" , '''2''' , Springer  (1982)  pp. §1</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Yamada,   "The Schur subgroup of the Brauer group" , Springer  (1974)</TD></TR></table>