arXiv:2611.15432 · v1

Weak Unit Reducibility of the Eleventh Cyclotomic Field

首次提交 2026-11-11 21:00

版本更新 2026-11-11 21:00

雷达观测 2026-11-13 11:11

摘要

We prove that the cyclotomic field $K_{11}=\mathbb{Q}(ζ_{11})$ is unit reducible but not strongly unit reducible. So every nonzero principal fractional ideal of $K_{11}$ has a generator that is a shortest nonzero vector. For $I=h\mathcal{O}_{K_{11}}$, every shortest nonzero vector $α\in I$ satisfies $N_{K_{11}/\mathbb{Q}}(α/h)\in\{1,11\}$. The proof uses Sigrist's Hermite bound for cyclic forms and an averaging identity for cyclotomic units. For principal ideals in $K_{11}$, ideal-SVP and the shortest-generator problem are polynomial-time equivalent.