Computation of the Euclidean minimum of algebraic number fields

Abstract

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri in 2007. With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 8 in any signature, especially for cyclotomic fields, and many new examples of norm-Euclidean or non-norm-Euclidean algebraic number fields. Then, we show how to apply the algorithm to study extensions of norm-Euclideanity.

The Euclidean algorithm in quintic and septic cyclic fields

“…Up until this point, it seems that nothing was known about the remaining fields in the table. We use the algorithm of Cerri from [3] (which has recently been extended by the first author in [11]) with some additional modifications to show that all five fields with ℓ = 5, 7 in Table 1.1 are norm-Euclidean. In fact, we compute the Euclidean minimum…”

“…Previously, Cerri computed that the Euclidean minimum of the cyclic quintic field with conductor f = 11 is equal to 1/11 (see [3]). We compute the Euclidean Table 1.1, using the algorithm described in [11].…”

Computation of Euclidean minima in totally definite quaternion fields

“…. , n} such that |σ i (ε)| > 1, and by i) (with p = 0) we have (11) x i+kn = t i+kn for every k ∈ {0, 1, 2, 3}.…”

“…We will answer to this question in Section 2, showing that the equality is always satisfied and that property (P ) holds when K = Q. Showing that it also holds when K is quadratic seems out of reach, as in the number field case, which is a famous conjecture due to Barnes and Swinnerton-Dyer. As in the number field case (see [4,11]), it is also natural to ask whether it is possible to use an algorithm allowing to compute M (Λ) = M (Λ) and to check that property (P ) is satisfied, even in the conjectural case. We will see that such an algorithm is already well known when K = Q so that we have just to study the case where K has degree at least 2.…”

Examples of Norm-Euclidean Ideal Classes

“…We can modify the algorithm described in [12] In practice, this algorithm is almost always successful, its execution may even be shorter if we only want to know if M (K, [I]) < 1.…”