No plane Keller map has prime field degree

Moskowicz's preprint proves that the plane conjecture holds for every Keller map whose field extension has prime degree.^[1] The proof splits on whether $xy$ lies in $\mathbb C(p,q)$. In one case every monomial generates the extension, which forces degree 2. In the other, restriction to a line gives injectivity. The argument ends by forcing degree 1 or 2. It then applies Keller's birational theorem, which closes degree 1. For degree 2 it uses the Galois criterion, which closes degree 2 because a quadratic extension is Galois, invoked through Bass–Connell–Wright's Theorem 2.12.

The atlas marks this V4, as a claim. It is one unrefereed preprint, and the research found no independent check or published critique. The refereed record in the plane excludes degrees 3 and 4 (Orevkov; Domrina), and every degree up to 5 (Żołądek).

In three dimensions the obstruction fails: Gallagher's family realizes every prime from 3. This result bounds the field degree, which is a different quantity from the component degrees in the gcd constraints.

1. ↑ V. Moskowicz, There are no Keller maps having prime degree field extensions, arXiv:2407.13795 (16 July 2024; one version, eight pages, no journal reference).