Undecidability of infinite towers of Kummer extensions of $\mathbb{F}_p(t)$

TL;DR

This paper demonstrates the undecidability of certain infinite field extensions related to Kummer extensions over function fields, assuming resolution of singularities, by combining algebraic and model-theoretic techniques.

Contribution

It establishes the undecidability of the first-order theory of specific infinite Kummer extension fields over finite fields, under a key geometric assumption.

Findings

Undecidability of the theory of compositum fields with roots of polynomials

Application of algebraic and model-theoretic methods to positive characteristic fields

Conditional results assuming resolution of singularities

Abstract

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory and builds on work of Anscombe, Dittmann and Fehm. As an application, we show that, for each finite field $\mathbb{F}$ of odd characteristic and any positive integer $n$ coprime with the characteristic of $\mathbb{F}$, the first-order theory of the field given by the compositum of the fields generated by adjoining the $n$--th roots of all monic irreducible polynomials in $\mathbb{F}[t]$, of degree divisible by $n$ is undecidable in the language of rings with the variable $t$ as a constant.