Transfinite Galois Theory

TL;DR

This paper extends Galois theory to non-Archimedean ordered fields using naked polynomials, establishing a transfinite Galois framework that generalizes classical algebraic methods.

Contribution

It introduces the concept of naked polynomials over non-Archimedean fields and shows their polynomial rings form Euclidean domains, enabling a transfinite Galois theory.

Findings

Naked polynomial rings are Euclidean domains.

Generalization of Galois theory to non-Archimedean fields.

Lifting of splitting and algebraic closure processes.

Abstract

In this paper I generalize the notion of a polynomial over an ordered field to that of a naked polynomial over a non-Archimedean ordered field, subsequently showing that the notion of a naked polynomial ring forms an Euclidean domain. This canonically generalizes the methods of Galois theory of fields and polynomial rings to a transfinite Galois theory of non-Archimedean ordered fields and naked polynomial rings, lifting the processes of splitting and algebraic closure to non-Archimedean ordered fields.