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

Graves, Hester; Murty, M.

doi:10.1090/S0002-9939-2013-11602-3

A family of number fields with unit rank at least $4$ that has Euclidean ideals
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by Hester Graves and M. Ram Murty PDF

Proc. Amer. Math. Soc. 141 (2013), 2979-2990 Request permission

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 $\mathbb {Q}$ 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 $\mathbb {Q}(\sqrt {5}, \sqrt {21}, \sqrt {22})$ has such an ideal class.

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