Difference between revisions

prime-degree-plane

(page created)

Revision 276 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__prime-degree-plane.md).”

Status and infobox

status: — → proved
tier: — → V4
kind: — → theorem
year: — → 2024
statement: — → If $p,q\in\mathbb C[x,y]$ have constant nonzero Jacobian, then $[\mathbb C(x,y):\mathbb C(p,q)]$ is not a prime number.
short: — → [ℂ(x,y) : ℂ(p,q)] is never prime for a Keller map (unrefereed)

Relations

+ special case of jc-plane: Moskowicz's preprint proves that the plane conjecture holds for every Keller map whose field extension has prime degree. The proof splits on whether $xy$ lies in $\mathbb C(p,q)$ .
+ uses keller-birational: It then applies Keller's birational theorem, which closes degree 1.
+ uses campbell-galois: 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.

Source +17 −0

+---

+title: No plane Keller map has prime field degree

+short: [ℂ(x,y) : ℂ(p,q)] is never prime for a Keller map (unrefereed)

+kind: theorem

+status: proved

+tier: V4

+year: 2024

+authors: Vered Moskowicz

+statement: If $p,q\in\mathbb C[x,y]$ have constant nonzero Jacobian, then $[\mathbb C(x,y):\mathbb C(p,q)]$ is not a prime number.

+---

+Moskowicz's preprint proves that [[special-case-of::Statement:jc-plane|the plane conjecture holds for every Keller map whose field extension has prime degree]].<ref>V. Moskowicz, [There are no Keller maps having prime degree field extensions](https://arxiv.org/abs/2407.13795), arXiv:2407.13795 (16 July 2024; one version, eight pages, no journal reference).</ref> 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 [[uses::Statement:keller-birational|Keller's birational theorem, which closes degree 1]]. For degree 2 it uses [[uses::Statement:campbell-galois|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: [[Statement:gallagher-families|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 [[Statement:plane-gcd-bounds|gcd constraints]].

+<references />

Revision 276 as rendered

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.

↑ 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).