Difference between revisions of "Field generated by character values"

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(Created page with "==Definition== Suppose <math>G</math> is a finite group. Pick a characteristic that is either zero or a prime not dividing the order of <math>G</math>. The '''field generate...")
 
 
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==Facts==
 
==Facts==
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===Relationship with cyclotomic extensions===
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* In characteristic zero, [[field generated by character values is contained in a cyclotomic extension of rationals]], because [[characters are cyclotomic integers]].
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* [[Field generated by character values need not be cyclotomic]]
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===Uniqueness and relationship with splitting fields===
  
 
* The field generated by character values is unique up to isomorphism of fields.
 
* The field generated by character values is unique up to isomorphism of fields.
 
* The field generated by character values is contained in every splitting field, and hence also in every [[minimal splitting field]].
 
* The field generated by character values is contained in every splitting field, and hence also in every [[minimal splitting field]].
* The field generated by character values is a cyclotomic extension of the rationals (in characteristic zero) because [[characters are cyclotomic integers]].
 
 
* [[Field generated by character values is splitting field implies it is the unique minimal splitting field]]
 
* [[Field generated by character values is splitting field implies it is the unique minimal splitting field]]
 
* [[Field generated by character values need not be a splitting field]]
 
* [[Field generated by character values need not be a splitting field]]

Latest revision as of 00:55, 1 February 2013

Definition

Suppose G is a finite group. Pick a characteristic that is either zero or a prime not dividing the order of G. The field generated by character values for G in that characteristic is the smallest field in that characteristic containing the values of all the characters of irreducible representations of G over a splitting field in that characteristic.

Facts

Relationship with cyclotomic extensions

Uniqueness and relationship with splitting fields