Certain monomial characters of \(p'\)-degree. (Q1925814)

Let \(G\) be a \(p\)-solvable finite group, let \(P\) be a Sylow \(p\)-subgroup of \(G\), and assume that \(N_G(P)/P\) has odd order. Let \(\mathbb Q_{p^a}\) be the field of rational numbers extended by a primitive \(p^a\)-th root of one. Let \(\chi\) be an irreducible character of \(G\) whose degree is not divisible by \(p\), and whose values lie in \(\mathbb Q_{p^a}\). It is proved in the paper that \(\chi\) is then induced from a linear character of some subgroup of \(G\) whose values are in \(\mathbb Q_{p^a}\). Furthermore, this linear character is unique up to \(G\)-conjugacy.