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

## Statement

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

Equivalently, 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.

## Related facts

### Similar facts

- Sufficiently large implies splitting: This in particular shows that there exists
*at least one*minimal splitting field that is contained in a cyclotomic extension of rationals. - Field generated by character values is splitting field implies it is the unique minimal splitting field

### Opposite facts

- Field generated by character values need not be cyclotomic
- Minimal splitting field need not be unique
- Splitting not implies sufficiently large
- Field generated by character values is splitting field implies it is the unique minimal splitting field
- Minimal splitting field need not be cyclotomic
- Minimal splitting field need not be contained in a cyclotomic extension of rationals

## Facts used

## Proof

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 roots of unity where is the exponent of . 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.