Splitting fields of real irreducible representations of finite groups

Abstract

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb{Q}(\zeta_n)$ to one over $E$.