  When $G$ is solvable group, we prove that the number of conjugacy classes of
elements of prime power order is less than or equal to the number of
irreducible characters with values in fields where $\mathbb {Q}$ is extended by
prime power roots of unity. We then combine this result with a theorem of
H\'ethelyi and K\"ulshammer that bounds the order of a finite group in terms of
the number of conjugacy classes of elements of prime power order to bound the
order of a solvable group by the number of irreducible characters with values
in fields extended by prime power roots of unity. This yields for solvable
groups a generalization of Landau's theorem.
