Existentially generated subfields of large fields

TL;DR

This paper investigates subfields of large fields generated by infinite existentially definable sets, showing such subfields contain certain power subfields, thus extending previous results and using model-theoretic methods involving henselian fields.

Contribution

It generalizes Fehm's result by demonstrating that existentially generated subfields of large fields contain specific power subfields, using a novel approach involving henselian fields.

Findings

Existentially generated subfields contain p^n-th power subfields for some n.

The method involves studying existentially generated subfields of henselian fields.

The results extend previous work by removing the perfectness assumption.

Abstract

We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated. Let $L$ be a large field of characteristic exponent $p$, and let $E\subseteq L$ be an infinite existentially generated subfield. We show that $E$ contains $L^{(p^{n})}$, the $p^{n}$-th powers in $L$, for some $n<\omega$. This generalises a result of Fehm, which shows $E=L$ under the assumption that $L$ is perfect. Our method is to first study existentially generated subfields of henselian fields. Since $L$ is existentially closed in the henselian field $L((t))$, our result follows.