  Motivated by the behavior of the trace pairing over tame cyclic number
fields, we introduce the notion of tame lattices. Given an arbitrary
non-trivial lattice $\mathcal{L}$ we construct a parametric family of full-rank
sub-lattices $\{\mathcal{L}_{\alpha}\}$ of $\mathcal{L}$ such that whenever
$\mathcal{L}$ is tame each $\mathcal{L}_{\alpha}$ has a basis of minimal
vectors. Furthermore, for each $\mathcal{L}_{\alpha}$ in the family a basis of
minimal vectors is explicitly constructed.
