Vol. 51, No. 2, 1974

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Minimal splitting fields for group representations

Burton I. Fein

Vol. 51 (1974), No. 2, 427–431

Let T be a complex irreducible representation of a finite group G of order n and let χ be the character afforded by T. An algebraic number field K Q(χ) is a splitting field for χ if T can be written in K. The minimum of [K : Q(χ)], taken over all splitting fields K of χ, is the Schur index mQ(χ) of χ. In view of the famous theorem of R. Brauer that Q(e2πi∕n) is a splitting field for χ, it is natural to ask whether there exists a splitting field L with Q(e2tti∕n) L Q(χ) and [L : Q(χ)] = mQ(χ). In this paper examples are constructed which show that such a splitting field L does not always exist. Sufficient conditions are also obtained which guarantee the existence of a splitting field L as above.

Mathematical Subject Classification 2000
Primary: 20C15
Received: 15 October 1973
Published: 1 April 1974
Burton I. Fein