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

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


 * 1) The number of irreducible representations of $$G$$ 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.
 * 2) The number of conjugacy classes in $$G$$ of rational elements, i.e., elements that are conjugate to any other element that generates the same cyclic subgroup.

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