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On the Northcott property for infinite extensions

Martin Widmer

Vol. 2 (2023), No. 1, 1–14
Abstract

We start with a brief survey on the Northcott property for subfields of the algebraic numbers ¯. Then we introduce a new criterion for its validity (refining the author’s previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier’s theorem, stating that the maximal abelian extension of a number field K contained in K(d) has the Northcott property, follows very easily from this refined criterion. Here K(d) denotes the composite field of all extensions of K of degree at most d.

Keywords
Weil height, Northcott property, property (N), Northcott's theorem, abelian extensions, Silverman's inequality
Mathematical Subject Classification
Primary: 11G50, 11R04
Secondary: 11R06, 11R20, 37P30
Milestones
Received: 31 May 2022
Revised: 27 September 2023
Accepted: 5 December 2023
Published: 31 December 2023
Authors
Martin Widmer
Department of Mathematics
Royal Holloway, University of London
Egham
United Kingdom