On all numbers great and small (Topological fields of Conway's numbers and their completions)

TL;DR

This paper explores the topological and algebraic properties of Conway's class of numbers, constructing completions of subfields and establishing fundamental algebraic results within these extended number systems.

Contribution

It introduces a method to construct completions of subfields of Conway's numbers and proves key algebraic properties like root existence and algebraic closure in these extended fields.

Findings

Every positive element in the constructed fields has a unique n-th root.

Odd-degree polynomials over these fields have roots within the fields.

The complexification of these fields is algebraically closed.

Abstract

The proper Class $\bf{No}$ of all Conway's numbers $\cite{l3}$ is considered as a region of investigation. It turns out to be a total ordered Field (i.e., a field whose domain is a proper Class) and this totally, or linear ordered Class, containing the real numbers ${\mathbb R}$ and the ordinal numbers {\bf On}. For any subfield $F$ of $\bf{No}$, i.e., $F$ is a set nor proper class, considered with topology induced by a linear ordering on $F$ a completion $\tilde F$ is constructed; in particular, for $\zeta=\omega^{\omega^\mu}$, $0\leq\mu<\Omega$, and for a specially defined subfield $F={\mathbb P}_\zeta\subset{\bf No}$ a complete subfield ${\mathbb R}_\zeta\subset{\bf No}$ is defined as $\tilde {\mathbb P}_\zeta$. Fundamental (Cauchy) sequences $(x_\alpha)_{0\leq\alpha<\zeta}$ are considered in a subfield $F\subset {\mathbb P}_\zeta\subset{\bf No}$, where $\zeta$ is the smallest ordinal number which does not belong to $F$, and they are the main instrument in the paper. A fragment of Mathematical Analysis in ${\mathbb R}_\zeta$ is given and two of its non-trivial results are presented: every positive number $x\in{\mathbb R}_\zeta$ has a unique $n$-th root in ${\mathbb R}_\zeta$, for each positive integer $n$ and every odd-degree polynomial with coefficients in ${\mathbb R}_\zeta$ has a root in ${\mathbb R}_\zeta$. Hence so-called fundamental theorem of algebra: the ring ${\mathbb R}_\zeta[i]\stackrel{def}{=}{\mathbb C}_\zeta$ of all numbers of the form $x+iy$ ($x,y\in{\mathbb R}_\zeta$), $i^2=-1$, is an algebraically closed field.