On fake subfields of number fields

TL;DR

This paper explores the phenomenon of 'fake subfields' in number fields, where local prime splitting data suggests containment but globally no such subfield exists, revealing failures in local-global principles.

Contribution

The authors systematically construct examples of fake subfields not arising from arithmetic equivalence, highlighting new failures of local-global principles in number field containment.

Findings

Existence of fake subfields not explained by arithmetic equivalence

Construction methods for fake subfields provided

Failure of local-global principle related to zeta functions

Abstract

We investigate the failure of a local-global principle with regard to "containment of number fields"; i.e., we are interested in pairs of number fields $(K_1,K_2)$ such that $K_2$ is not a subfield of any algebraic conjugate $K_1^\sigma$ of $K_1$, but the splitting type of any single rational prime $p$ unramified in $K_1$ and in $K_2$ is such that it cannot rule out the containment $K_2\subseteq K_1^\sigma$. Examples of such situations arise naturally, but not exclusively, via the well-studied concept of arithmetically equivalent number fields. We give some systematic constructions yielding "fake subfields" (in the above sense) which are not induced by arithmetic equivalence. This may also be interpreted as a failure of a certain local-global principle related to zeta functions of number fields.