Minkowski shapes of pure number fields

Abstract

We study the Minkowski shape of pure number fields \[ K_a=\mathbb Q(θ),\qquad θ^n=a. \] For admissible parameters satisfying an explicit local hypothesis at the primes dividing $n$, we prove a discrete--archimedean factorization \[ \mathrm{sh}(K_a)=\bigl[C(a)^{\mathsf T}\mathrm{diag}\bigl(s_1(a),\dots,s_{n-1}(a)\bigr)C(a)\bigr], \] where the $s_m(a)$ arise from normalized monomials and $C(a)\in\mathrm{GL}_{n-1}(\mathbb Q)$ comes from a normalized integral basis. This yields a uniform odd/even rigidity dichotomy: for every odd $n\geq 3$, the Minkowski shape is a complete invariant among admissible pure degree-$n$ fields, whereas for $n=2r$ it determines the core field $\mathbb Q(|a|^{1/r})$; on the squarefree admissible subfamily it is complete up to sign, although infinitely many non-isomorphic pairs $K_a$ and $K_{-a}$ have the same shape. We also derive explicit formulas for $|\mathrm{disc}(K_a)|$, including exponent-vector and divisor-lattice factorizations. Finally, we show that the pure-field shape locus is supported on rational diagonal leaves in shape space: unconditionally it lies in a countable union of closed leaves, while under the same local hypothesis only finitely many leaves occur in each fixed degree. On a fixed normalized stratum, the shape depends only on ratio variables, whereas discriminant growth is governed by independent product variables.