# Construction of the Lindstr\"om valuation of an algebraic extension

## TL;DR

This paper provides a direct construction of the Lindström valuation for algebraic matroids of field extensions, clarifying its structure using inseparable extension theory.

## Contribution

It introduces a new direct method to construct the Lindström valuation, bypassing the matroid flock approach, and describes its valuation, circuits, and cocircuits.

## Key findings

- Explicit description of the Lindström valuation
- Characterization of valuated circuits and cocircuits
- Connection to inseparable field extensions

## Abstract

Recently, Bollen, Draisma, and Pendavingh have introduced the Lindstr\"om valuation on the algebraic matroid of a field extension of characteristic p. Their construction passes through what they call a matroid flock and builds on some of the associated theory of matroid flocks which they develop. In this paper, we give a direct construction of the Lindstr\"om valuated matroid using the theory of inseparable field extensions. In particular, we give a description of the valuation, the valuated circuits, and the valuated cocircuits.