On possible values of the $m$-invariant

TL;DR

This paper proves that all positive integers except 3 and 5 can be realized as the $m$-invariant of some field, expanding understanding of the possible values of this invariant.

Contribution

It establishes the complete set of positive integers that can serve as the $m$-invariant, except for 3 and 5, providing a comprehensive classification.

Findings

Every positive integer except 3 and 5 is realizable as an $m$-invariant.

The $m$-invariant can take on almost all positive integer values.

The exceptions 3 and 5 are proven to be impossible as $m$-invariants.

Abstract

We show that every positive integer different from $3$ and $5$ can be realized as the $m$-invariant of a field.