 In particular this applies to the case of totally real quadratic number fields with trivial class group.In addition (and without assuming ERH), if the field "K" is a Galois extension of Q, has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean.An immediate corollary of this is that if the number field is Galois over Q, its class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.