Schur indices and sums of squares

Fein, Burton

doi:10.1090/S0002-9939-1975-0374249-6

Schur indices and sums of squares
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Proc. Amer. Math. Soc. 51 (1975), 31-34 Request permission

Abstract:

Let $G$ be a finite group of exponent $n, F$ a field of characteristic zero, $\varepsilon$ a primitive $n$th root of unity, and suppose that the Sylow $2$-subgroup of the Galois group of $F(\varepsilon )$ over $F$ is cyclic. Let $\chi$ be an absolutely irreducible character of $G$. Strengthening a recent result of Goldschmidt and Isaacs, it is shown that if — 1 is a sum of two squares in $F$, then the Schur index of $\chi$ over $F$ is odd.

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