Difference between revisions of "Field generated by character values"
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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]]. | ||
| + | * [[Field generated by character values need not be cyclotomic]] | ||
| + | |||
| + | ===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]]. | ||
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* [[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
Contents
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
Relationship with cyclotomic extensions
- In characteristic zero, field generated by character values is contained in a cyclotomic extension of rationals, because characters are cyclotomic integers.
- Field generated by character values need not be cyclotomic
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 contained in every splitting field, and hence also in every 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
