Polynomial composites and certain types of fields extensions

TL;DR

This paper explores the relationship between polynomial composites with coefficients from a field extension and the characterization of certain field extensions, aiming to understand ideals in these composites.

Contribution

It provides new characterizations of some known field extensions using polynomial composites and addresses the open problem of ideal characterization in this context.

Findings

Characterization of certain field extensions via polynomial composites

Identification of properties linking composites and field extensions

Open problem posed for ideal characterization in polynomial composites

Abstract

In this paper I consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. I present the characterization of some known field extensions in terms of polynomial composites. This paper contains the opening problem of characterization of ideals in polynomial composites with respect to various field extensions. I also present the full possible characterization of certain field extensions.