Explicit Euclidean division algorithms for some degree 8 number rings

Abstract

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.