On Maximal and Minimal Linear Matching Property

TL;DR

This paper introduces minimal and maximal linear matching properties for field extensions, proving that non-algebraically closed fields have minimal property and algebraic number fields have maximal property, along with a new proof of the fundamental theorem of algebra.

Contribution

It defines new linear matching properties for field extensions and establishes their presence in specific classes of fields, providing new insights and proofs.

Findings

Non-algebraically closed fields have minimal linear matching property.

Algebraic number fields have maximal linear matching property.

A shorter proof of the fundamental theorem of algebra is provided.

Abstract

The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if K is not algebraically closed, then K has minimal linear matching property. In this paper we will prove that algebraic number fields have maximal linear matching property. We also give a shorter proof of a result established in [6] on the fundamental theorem of algebra.