A characterization property on field equivalent to algebraicity on Banach spaces

TL;DR

This paper introduces a logical property characterizing algebraicity in fields that are Banach spaces, extending Tyszka's work from real and p-adic fields to all characteristic zero fields and providing weaker results for positive characteristic fields.

Contribution

It presents a new property equivalent to algebraicity in characteristic zero fields that are Banach spaces and establishes a weaker equivalence for positive characteristic fields.

Findings

Property is equivalent to algebraicity in characteristic zero Banach fields.

Weaker equivalence established for positive characteristic Banach fields.

Abstract

In his article "A discrete form of the theorem that each field endomorphism of $\mathbb{R}$ ($\mathbb{Q}_p$) is the identity", Tyszka introduce a logical property which is equivalent to algebraicity in $\mathbb{R}$ and in $\mathbb{Q}_p$. Amazingly, the property is no longer equivalent to algebraicity in $\mathbb{C}$. This article present a similirar property which is equivalent to algebraicity in any field of characteristic zero which is also a Banach space, and prove a weaker equivalency for fields of positive charcteristic (which are also Banach spaces).