  We give an explicit and character-free construction of a complete set of
orthogonal primitive idempotents of a rational group algebra of a finite
nilpotent group and a full description of the Wedderburn decomposition of such
algebras. An immediate consequence is a well-known result of Roquette on the
Schur indices of the simple components of group algebras of finite nilpotent
groups. As an application, we obtain that the unit group of the integral group
ring $\Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that
is generated by three nilpotent groups for which we have an explicit
description of their generators. Another application is a new construction of
free subgroups in the unit group. In all the constructions dealt with, pairs of
subgroups $(H,K)$, called strong Shoda pairs, and explicit constructed central
elements $e(G,H,K)$ play a crucial role. For arbitrary finite groups we prove
that the primitive central idempotents of the rational group algebras are
rational linear combinations of such $e(G,H,K)$, with $(H,K)$ strong Shoda
pairs in subgroups of $G$.
