JOURNAL OF ALGEBRA 68, 75-78 (1981) Character Values of Groups of Odd Order and a Question of Feit R. Gow Mathematics Department, University Cortege Dublin, Belfeld Dublin 4, Ireland Communicated by W. Feit Received March 3 1, 1980 Let x be an irreducible complex character of a finite group G and let Q&) denote the field obtained by adjoining the values of x to the rational field Q. Let Q, denote the field obtained by adjoining a primitive mth root of unity to Q. We say that x requires mth roots of unity if Qk) is contained in Q, and m is the smallest positive integer with this property. The following open question is raised by Feit 12, p. 411. Suppose that x requires mth roots of unity. Is it true that G contains an element of order m? The purpose of this paper is to provide an affirmative answer to the question in the case that G has odd order. In ( 11, Brauer showed that there is an affirmative answer to the question without restriction on the group order provided that each prime divisor of m occurs to the second power or more. We simply continue Brauer’s argument to deal with the prime divisors that occur only to the first power. Using the fact that no nonidentity element can be conjugate to its inverse in a group of odd order, we are able to avoid difficulties that seem to obstruct a general proof. The final section contains a partial converse to the main theorem. 1. PRELIMINARIES Let p be a prime divisor of j G / and let ]G / = n = hp*, where (h,p) = 1. We say that an irreducible character x of G is p-rational if Q@) is contained in Q,,. Thus if x requires mth roots of unity, it is not p-rational for any prime divisor p of m. The Galois group H of Q, over Q,, is isomorphic to the Galois group of QPa over Q and is thus cyclic if p is odd. It is well known that actions of H on the irreducible characters and classes of G can be defined and that these actions are permutation isomorphic when H is cyclic. This is the substance of Brauer’s permutation lemma 12, p. 661. The following easily proved lemma appears in [3]. 75 002 l-8693/8 1 /O 10075-04S02.00,‘0 Copyrtght C 1981 b) Academic Press, Inc. All rlghls of reproducuon m any form reserved.