Additive Polynomials and Their Role in the Model Theory of Valued Fields

TL;DR

This paper explores the significance of additive and p-polynomials in the model theory of valued fields of positive characteristic, establishing foundational properties and structural results about their extensions and modules.

Contribution

It introduces new structural insights into valued fields using additive polynomials, including the existence of Frobenius-closed bases and characterization of wild extensions.

Findings

Existence of Frobenius-closed bases in algebraic function fields.

Valued fields of positive characteristic are free modules over rings of additive polynomials.

Minimal purely wild extensions are generated by p-polynomials.

Abstract

We discuss the role of additive polynomials and $p$-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields $F|K$ in one variable and deduce that $F/K$ is a free module over the ring of additive polynomials with coefficients in $K$. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a $p$-polynomial.