Topological Transcendental Fields

TL;DR

This paper explores the structure and classification of topological transcendental fields within the complex numbers, revealing their topological properties and the vast diversity of such fields based on transcendental sets.

Contribution

It introduces the concept of topological transcendental fields, classifies their topological types, and demonstrates the existence of many non-homeomorphic fields distinguished by their transcendental sets.

Findings

There are exactly 2^{ ext{aleph}_0} countably infinite topological transcendental fields.

Each such field is homeomorphic to the rational numbers with the usual topology.

There are 2^{2^{ ext{aleph}_0}} non-homeomorphic topological transcendental fields of the form 5(T) with T as Liouville numbers.

Abstract

This article initiates the study of topological transcendental fields $\FF$ which are subfields of the topological field $\CC$ of all complex numbers such that $\FF$ consists of only rational numbers and a nonempty set of transcendental numbers. $\FF$, with the topology it inherits as a subspace of $\CC$, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is $\QQ(T)$, the extension of the field of rational numbers by a set $T$ of transcendental numbers. It is proved that there exist precisely $2^{\aleph_0}$ countably infinite topological transcendental fields and each is homeomorphic to the space $\QQ$ of rational numbers with its usual topology. It is also shown that there is a class of $2^{2^{\aleph_0} }$ of topological transcendental fields of the form $\QQ(T)$ with $T$ a set of Liouville numbers, no two of which are homeomorphic.