Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy

Statement

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

1. The number of irreducible linear representations of $G$ over the field of rational numbers.
2. The number of linear representations of $G$ over $\mathbb{C}$ with rational character values and for which no proper nonzero subrepresentation has rational character values.
3. The number of equivalence classes in $G$ under rational conjugacy.
4. The number of conjugacy classes of cyclic subgroups in $G$.

Note that the result is not true for arbitrary finite groups.

Caveats

The number of irreducible representations over rationals is not the same as the number of irreducible representations over the complex numbers that can be realized over the rationals. The latter number is either smaller or equal, and it is equal when the group is a rational group, which means that any two elements generating the same cyclic subgroup are conjugate.

Also, although the counts in (1) and (2) are equal, it is possible for a rational character to arise from an irreducible representation over the complex numbers that is not realized over the rationals. However, some multiple of that representation can be realized over the reals. This explains the equality of counts in (1) and (2). The smallest multiple used is termed the Schur index.