On the Northcott property for infinite extensions

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We start with a brief survey on the Northcott property for subfields of the algebraic numbers $\Qbar$. 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 $d$.
Original languageEnglish
PublisherEssential Number Theory
Publication statusPublished - 17 Oct 2023


  • math.NT
  • Primary 11R04, 11G50 Secondary 11R06, 11R20, 37P30

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