A family of number fields with unit rank at least 4 that has Euclidean ideals

Abstract

We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over Q \mathbb {Q} is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that Q ( 5 , 21 , 22 ) \mathbb {Q}(\sqrt {5}, \sqrt {21}, \sqrt {22}) has such an ideal class.