Ring generated by character values: Difference between revisions
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Note that in prime characteristic, it is the same as the [[field generated by character values]]. | Note that in prime characteristic, it is the same as the [[field generated by character values]]. | ||
In characteristic zero, it is a cyclotomic extension of the ring of integers <math>\mathbb{Z}</math>, because [[characters are cyclotomic integers]]. | In characteristic zero, it is contained in a cyclotomic extension of the ring of integers <math>\mathbb{Z}</math>, because [[characters are cyclotomic integers]]. | ||
Note that this differes from the [[character ring]], which is the ring of characters as functions. | Note that this differes from the [[character ring]], which is the ring of characters as functions. | ||
Latest revision as of 20:55, 13 July 2011
Definition
In characteristic zero or a prime characteristic
Suppose is a finite group and fix a characteristic that is either zero or a prime not dividing the order of . The ring generated by character values is the smallest ring containing all the character values of over a splitting field.
Note that in prime characteristic, it is the same as the field generated by character values.
In characteristic zero, it is contained in a cyclotomic extension of the ring of integers , because characters are cyclotomic integers.
Note that this differes from the character ring, which is the ring of characters as functions.