PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 10, October 2006, Pages 2833–2837 S 0002-9939(06)08564-9 Article electronically published on April 11, 2006 CHARACTERS OF p ′ -DEGREE WITH CYCLOTOMIC FIELD OF VALUES GABRIEL NAVARRO AND PHAM HUU TIEP (Communicated by Jonathan I. Hall) Abstract. If p is a prime number and G is a finite group, we show that G has an irreducible complex character of degree not divisible by p with values in the cyclotomic field Q p . 1. Introduction R. Gow conjectured that every finite group of even order has a nontrivial irre- ducible complex character with odd degree and rational values. This conjecture was finally proven in [7]. In this note we seek an analog of this result which works for every prime p. If G is a finite group and χ ∈ Irr(G) is an irreducible complex character of G, we denote by Q(χ) the field of values of χ. Also, we let Q n be the cyclotomic field generated by a primitive nth root of unity. Theorem A. Let p be a prime and let G be a finite group of order divisible by p. Then there exists a nontrivial χ ∈ Irr(G) of p ′ -degree such that Q(χ) ⊆ Q p . Of course, the p = 2 case of Theorem B is Gow’s conjecture. We are able, in fact, to prove the following. Theorem B. Let p be a prime and let G be a finite group. Then the trivial character is the only p ′ -degree irreducible character of G with values in Q p if and only if G is a group of odd order not divisible by p. Once we have that groups of order divisible by p have nontrivial irreducible p ′ -degree characters with values in Q p , it is natural to ask if we can even obtain them having rational values. Of course, some conditions are necessary (since odd p-groups certainly do not possess these characters, nor does L 2 (3 2a+1 ) for p = 3). The following improves on some of the results in [5]. Theorem C. Let G be a finite group and let p be a prime. (i) Let G be nonsolvable. Assume that either p =3, or that p =3 and G has no composition factor isomorphic to L 2 (3 2a+1 ) for any a ≥ 1. Then G has a nontrivial, rational, irreducible character of p ′ -degree. Received by the editors April 22, 2005. 2000 Mathematics Subject Classification. Primary 20C15. The first author was partially supported by the Ministerio de Educaci´on y Ciencia proyecto MTM2004-06067-C02-01. The second author gratefully acknowledges the support of the NSA (grant H98230-04-0066). c 2006 American Mathematical Society Reverts to public domain 28 years from publication 2833 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use