Bases of minimal vectors in tame lattices

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.