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It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and…

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the…

Let $K$ be a cyclic cubic field and $\mathcal{O}_K$ be its ring of integers. In this note we prove that all cyclic cubic number fields with conductors in the interval $ [73, 11971]$ and with class number one are Euclidean.

In this note we present algorithms for computing Euclidean minima of cubic number fields; in particular, we were able to find all norm-Euclidean cubic number fields with discriminants -999 < d < 10000.

In 1977, Lenstra provided a criterion for norm-Euclideanity of number fields and noted that this criterion becomes ineffective for number fields of large enough degrees under the Generalised Riemann Hypothesis (GRH) for the Dedekind…

We study Euclidean ideal classes in real biquadratic fields and obtain unconditional existence results via genus theory. Lenstra showed (assuming the Generalized Riemann Hypothesis) that a number field with unit rank at least one admits a…

We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each…

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified,…

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied…

In this note we present techniques to compute inhomogeneous minima of norm forms; as an application, we determine all norm-Euclidean complex bicyclic quartic number fields.

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that…

The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean…

The list of norm-Euclidean imaginary quadratic fields is known and finite. For each known case, we give a division algorithm that finds a remainder at distance less than the Euclidean minimum of the field.

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu = 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result is…

We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show…

Let $K$ be a field of degree $n$ and discriminant with absolute value $\Delta$. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of $K$…

We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…

The Euclidean quantum field theory for the fields $\phi_{\Delta x}(x)$, which depend on both the position $x$ and the resolution $\Delta x$, constructed in SIGMA 2 (2006), 046, hep-th/0604170, on the base of the continuous wavelet…

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.