This article focuses on some rings of integers of number fields which are known to be norm-Euclidean domains, but for which no explicit algorithm computing the Euclidean division has yet been studied or implemented. The rings of integers we are interested in were proven to be Euclidean by H.W. Lenstra, Jr in 1978; they include the $n$-th cyclotomic rings for $n=15,20,24$. We present an algorithm performing Euclidean division in these rings based on Lenstra's proof and a closest vector computation by Conway and Sloane, and study its complexity. We give a complete implementation of the algorithm in SageMath. We also estimate the size of the remainders obtained when computing Euclidean divisions with this algorithm.

翻译：本文聚焦于若干已知为范数欧几里得整环的数域整数环，但针对这些环尚未有显式计算欧几里得除法的算法被研究或实现。我们所关注的整数环由H.W. Lenstra, Jr于1978年证明为欧几里得环；其中包括$n=15,20,24$时的$n$次分圆环。我们提出一种基于Lenstra证明及Conway与Sloane最近向量计算的算法，用于在这些环中执行欧几里得除法，并分析其复杂度。我们在SageMath中给出了该算法的完整实现。同时，我们估算了使用该算法计算欧几里得除法时所得余数的大小范围。