# Degree of irreducible representation over field of characteristic coprime to order divides product of order and Euler totient function of exponent

## Statement

Suppose $G$ is a finite group, $k$ is a field whose characteristic is relatively prime to the order of $G$, and $\varphi$ is an irreducible representation of $G$ over $k$ (which need not be absolutely irreducible, i.e., it may split over some algebraic extension of $k$). Then, the degree of $\varphi$ divides the product of the order of $G$ and the Euler totient function of the exponent of $G$.

## Related facts

### Facts over a splitting field

We can obtain much better bounds on the degrees of irreducible representations if $K$ is a splitting field for $G$. Specifically:

## Facts used

1. Sufficiently large implies splitting: Any sufficiently large field for a finite group $G$, i.e., a field that contains all the $m^{th}$ roots of unity where $m$ is the exponent of $G$, is also a splitting field for $G$.
2. Degree of irreducible representation divides group order