On the Northcott property for infinite extensions

Abstract

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 $d$.

References in corpus (1)

• On certain infinite extensions of the rationals with Northcott property

Cited by in corpus (2)

• Universal quadratic forms and Northcott property of infinite number fields
• Most totally real fields do not have universal forms or Northcott property