# Characters are cyclotomic integers

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This page describes a useful fact in character theory/linear representation theory arising from rudimentary linear algebra
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## Statement

Let  be a finite group, and  be any field. Then, the character of any finite-dimensional linear representation of  over  takes, at every element of the group, a value that is a cyclotomic integer over the prime subfield of : in other words, it is in the subring generated by all the roots of unity.

(The statement, does not require the characteristic of  to not divide the order of the group).

## Related facts

### Applications

• Characters are algebraic integers: Any cyclotomic integer is an algebraic integer, so every character takes values in algebraic integers.
• If, over a field of characteristic zero, a character takes a rational value, that value must be an integer.

## Proof

Given: A finite group , field , a finite-dimensional linear representation  of  over .  is the character of .

To prove: For any ,  is a cyclotomic integer over the prime subfield of .

Proof:  is the sum of eigenvalues of , counted with multiplicity, in the algebraic closure of . Since  is finite,  has finite order, so  has finite order, and hence  satisfies a polynomial of the form . Hence, every eigenvalue of  is a  root of unity, so  is a sum of  roots of unity. Hence,  is a cyclotomic integer.