# Difference between revisions of "Field generated by character values"

From Groupprops

(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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* 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]]. | + | * The field generated by character values is contained in 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]] | ||

+ | * [[Field generated by character values need not be cyclotomic]] |

## Revision as of 00:51, 1 February 2013

## Definition

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

## Facts

- 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 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 need not be a splitting field
- Field generated by character values need not be cyclotomic