  We give an algebraic proof of a class number formula for dihedral extensions
of number fields of degree $2q$, where $q$ is any odd integer. Our formula
expresses the ratio of class numbers as a ratio of orders of cohomology groups
of units and recovers similar formulas which have appeared in the literature as
special cases. As a corollary of our main result we obtain explicit bounds on
the (finitely many) possible values which can occur as ratio of class numbers
in dihedral extensions. Such bounds are obtained by arithmetic means, without
resorting to deep integral representation theory.
