# Number of irreducible representations over complex numbers with rational character values equals number of conjugacy classes of rational elements for any finite group whose cyclotomic splitting field is a cyclic extension of the rationals

From Groupprops

## Statement

Suppose is a finite group whose cyclotomic splitting field is a cyclic extension of the rationals. Then, the following numbers are equal:

- The number of irreducible representations of over the complex numbers whose characters are rational-valued. Note that this includes both rational representations and representations realized over an extension that still have all their character values rational.
- The number of conjugacy classes in of rational elements, i.e., elements that are conjugate to any other element that generates the same cyclic subgroup.

## Related facts

### Opposite facts

The result is not valid for *all* finite groups: Number of irreducible representations over complex numbers with rational character values need not equal number of conjugacy classes of rational elements

### Similar facts

The following hold for *any* finite group:

- Number of irreducible representations equals number of conjugacy classes
- Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements
- Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy
- Number of irreducible representations over reals equals number of equivalence classes under real conjugacy
- Number of orbits of irreducible representations equals number of orbits under automorphism group