On fake subfields of number fields

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.