Field generated by character values is contained in a cyclotomic extension of rationals

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Suppose G is a finite group and K is the field generated by character values for G in characteristic zero. Then, K is contained in a cyclotomic extension of the rationals.

Equivalently, K is a finite abelian extension of the field of rational numbers. This equivalence follows from a deep result of Galois theory called the Kronecker-Weber theorem, though the direction of relevance to us here is straightforward.

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  1. Characters are cyclotomic integers


The proof essentially follows from Fact (1), and the observation that since the group is finite, there are only finitely many characters involved, so there is a finite cyclotomic extension containing them all.

In fact, we can use the extension taking all d^{th} roots of unity where d is the exponent of G. This is the unique smallest sufficiently large field. It is a much more nontrivial result that such a cyclotomic extension is also a splitting field, i.e., the representations themselves can be realized over such an extension. For that, see sufficiently large implies splitting.