On the field of character values of finite solvable groups (Q1109130)

The following is proved, using the theory of factorization of characters into p-special characters. Theorem: Let G be a finite solvable group and \(\chi\) an irreducible complex character of G. Then \(| {\mathbb{Q}}_{f(\chi)}: {\mathbb{Q}}(\chi)|\) divides \(\chi(1).\) (Definition of symbols: Let \(\chi\in Irr(G):\) \({\mathbb{Q}}(\chi):={\mathbb{Q}}(\chi (g_ 1),...,\chi (g_ t))\), where \(G=\{g_ 1,...,g_ t\}\), let f(\(\chi)\) be the smallest positive integer f for which \({\mathbb{Q}}(\chi)\subseteq {\mathbb{Q}}(1^{1/f}).)\)