### A Banach space determined by the Weil height

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This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, $\mathbb{Q}$, discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the embedding of...

We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree.

Let p be an odd prime and let c be an integer such that c>1 and c divides p-1. Let G be a metacyclic group of order pc and let k be a field such that pc is prime to the characteristic of k. Assume that k contains a primitive pcth root of unity. We first characterize the normal extensions L/k with Galois group isomorphic to G when p and c satisfy a certain condition. Then we apply our characterization to the case in which k is an algebraic number field with ring of integers ℴ, and, assuming some...

We generalize Dirichlet's S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a ℚ-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over ℚ retain their linear independence...

Let $K$ be an algebraic number field and ${\mathcal{O}}_{K}$ the ring of integers of $K$. In this paper, we prove an analogue of Voronoï’s theorem for ${\mathcal{O}}_{K}$-lattices and the finiteness of the number of similar isometry classes of perfect ${\mathcal{O}}_{K}$-lattices.

We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of $\mathbb{Q}$. In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a $p$-adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.

1. Introduction. Number fields with the same zeta function are said to be arithmetically equivalent. Arithmetically equivalent fields share much of the same properties; for example, they have the same degrees, discriminants, number of both real and complex valuations, and prime decomposition laws (over ℚ). They also have isomorphic unit groups and determine the same normal closure over ℚ [6]. Strangely enough, it has been shown (for example [4], or more recently [6] and [7]) that this does...