Hall $\pi$-subgroups and characters of ${\pi }^{\mathrm{\prime }}$-degree

We study the relationship between the existence of Hall $\pi$-subgroups and that of irreducible characters of ${\pi }^{\mathrm{\prime }}$-degree with prescribed fields of values in finite groups. This work extends a result of Navarro and Tiep from a single odd prime to multiple odd primes.