# Characters are cyclotomic integers

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

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

(The statement, does not require the characteristic of $k$ 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 $G$, field $k$, a finite-dimensional linear representation $(V,\rho)$ of $G$ over $k$. $\chi$ is the character of $\rho$.

To prove: For any $g$, $\chi(g)$ is a cyclotomic integer over the prime subfield of $k$.

Proof: $\chi(g)$ is the sum of eigenvalues of $\rho(g)$, counted with multiplicity, in the algebraic closure of $k$. Since $G$ is finite, $g$ has finite order, so $\rho(g)$ has finite order, and hence $\rho(g)$ satisfies a polynomial of the form $x^n - 1$. Hence, every eigenvalue of $\rho(g)$ is a $n^{th}$ root of unity, so $\chi(g)$ is a sum of $n^{th}$ roots of unity. Hence, $\chi(g)$ is a cyclotomic integer.