On finiteness of the numbers of Euclidean fields in some classes of number fields (Q799729)

An ideal class C of a number field K is called norm-Euclidean if there is an ideal \({\mathfrak a}\) such that for every \(\alpha \in K\) there exists a \(\beta \in {\mathfrak a}\) such that \(| N_ K(\alpha -\beta)| <N_ K {\mathfrak a}.\) A field is Euclidean if and only if there is one ideal class and that is norm-Euclidean. Let \(F_ 1^{(n)}\) be the set of cyclic fields of degree n such that every rational prime factor of the discriminant is totally ramified, let \(F_ 2^{(\ell)}\) be \(\{{\mathbb{Q}}(^{\ell}\sqrt{m};\quad m\in {\mathbb{Z}}\},\) and let \(F_ 3^{(d)}\) be the set of non-Galois cubic fields K such that the Galois closure of K/\({\mathbb{Q}}\) contains \({\mathbb{Q}}(\sqrt{d}).\) By means of estimates of character sums it is shown that \(F_ 1^{(n)}\) and \(F_ 3^{(d)}\) contain only finitely many fields with a norm- Euclidean ideal class: the same is true for \(F_ 2^{(\ell)}\) under the assumption of the Generalized Riemann Hypothesis. Moreover \(F_ 2^{(5)}\) contains only finitely many Euclidean fields, without assuming the G.R.H..