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 $E$.