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# A Curiosity Concerning the Degrees of the Characters of a Finite Group

K. L. Fields
Proceedings of the American Mathematical Society
Vol. 62, No. 1 (Jan., 1977), pp. 25-27
DOI: 10.2307/2041938
Stable URL: http://www.jstor.org/stable/2041938
Page Count: 3
Let $G$ be a finite group with irreducible characters $\{\ldots, \chi,\ldots\}$ and $K = \mathbf{Q}(\ldots, \chi, \ldots)$ the field generated over the rationals by their values. We will prove: $$\text{If}\quad K = \mathbf{Q} (\text{or if}\quad \lbrack K: \mathbf{Q} \rbrack\quad\text{is odd) then}\quad \prod_{\chi(1)\text{odd}} \chi(1) \text{is a perfect square}.$$