# Ring generated by character values

## 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.