There are no Keller maps having prime degree field extensions

TL;DR

This paper proves that no Keller map with an invertible Jacobian has a prime degree extension, advancing understanding of the Jacobian Conjecture in algebraic geometry.

Contribution

It establishes that Keller maps with prime degree field extensions do not exist, providing a new restriction on potential counterexamples to the Jacobian Conjecture.

Findings

No Keller map with prime degree extension exists.

Supports the conjecture that Keller maps are automorphisms.

Provides a new algebraic restriction on Keller maps.

Abstract

The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y) \mapsto (p,q) \in k[x,y]^2$ having an invertible Jacobian is an automorphism of $k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.