Further refinements of the Itô-Michler theorem are proved, with the p'-degree condition restricted to p-rational characters lying above the principal character of a Sylow p-subgroup (or strongly real characters when p=2).
Marinelli and Pham Huu Tiep, Zeros of real irreducible characters of finite groups, Algebra Number Theory 7 (2013), 567--593
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Degrees of $p$-rational characters and normality of Sylow $p$-subgroups
Further refinements of the Itô-Michler theorem are proved, with the p'-degree condition restricted to p-rational characters lying above the principal character of a Sylow p-subgroup (or strongly real characters when p=2).