Euclidean Totally Definite Quaternion Fields Over the Rational Field and Over Quadratic Number Fields

Abstract

totally definite quaternion fields over the rational field and over quadratic number fields. International Journal of Number Theory, World Scientific Publishing, 2013, 9 (3), pp.653-673. <10.1142/S1793042112501540>. <hal-00738164v2> EUCLIDEAN TOTALLY DEFINITE QUATERNION FIELDS OVER THE RATIONAL FIELD AND OVER QUADRATIC NUMBER FIELDSJEAN-PAUL CERRI, JÉRÔME CHAUBERT, AND PIERRE LEZOWSKI Abstract. In this article we study totally definite quaternion fields over the rational field and over quadratic number fields. We establish a complete list of all such fields which are Euclidean. Moreover, we prove that every field in this list is in fact norm-Euclidean. The proofs are both theoretical and algorithmic.

Totally indefinite Euclidean quaternion fields

“…Therefore, (3) shows that nrd F/K (a + bc p ) = trd F/K (a + bτ p ) mod p, which proves that nrd F/K (a + bc p ) / ∈ p, as expected.…”

“…Obviously, if nrd F/K (a) / ∈ p or trd F/K (a) / ∈ p, we may take, τ p = 0. Let us assume then that nrd F/K (a) ∈ p and trd F/K (a) ∈ p. Thanks to Lemma 2.2 (iii), there exists a 3 Let x, y be two elements of Z K . We say that x and y are coprime or that x is coprime to y when the ideals xZ K and yZ K are coprime.…”

“…We can define similarly left-Euclidean orders and left-norm-Euclidean orders by replacing bq by qb in (1) and (2). In fact, these two notions are equivalent, which allows to speak of Euclidean and norm-Euclidean orders (see [3]). Moreover, if F admits a Euclidean (repectively norm-Euclidean) order Λ, then Λ is maximal and every maximal order of F is also Euclidean (respectively norm-Euclidean), which enables us to speak of Euclidean (respectively norm-Euclidean) quaternion fields: quaternion fields admiting a Euclidean (respectively norm-Euclidean) maximal order.…”

“…Moreover, if F admits a Euclidean (repectively norm-Euclidean) order Λ, then Λ is maximal and every maximal order of F is also Euclidean (respectively norm-Euclidean), which enables us to speak of Euclidean (respectively norm-Euclidean) quaternion fields: quaternion fields admiting a Euclidean (respectively norm-Euclidean) maximal order. All these considerations are developed in [3] and will be recalled in Section 2.…”

Computation of Euclidean minima in totally definite quaternion fields

“…Note that these results still hold when F is not totally definite. For more details, the reader can refer to [8,2,6]. A first natural question arises: what can be said when the degree of K is 1 or 2, i.e.…”