# No plane Keller map has prime field degree

- kind: theorem
- status: proved (evidence tier V4)
- year: 2024
- authors: Vered Moskowicz
- in one line: [ℂ(x,y) : ℂ(p,q)] is never prime for a Keller map (unrefereed)

**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 [the plane conjecture holds for every Keller map whose field extension has prime degree](/wiki/Statement:jc-plane.md).[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](/wiki/Statement:keller-birational.md). For degree 2 it uses [the Galois criterion, which closes degree 2 because a quadratic extension is Galois](/wiki/Statement:campbell-galois.md), 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](/wiki/Statement:gallagher-families.md) realizes every prime from 3. This result bounds the field degree, which is a different quantity from the component degrees in the [gcd constraints](/wiki/Statement:plane-gcd-bounds.md).

## Relations

- special case of [jc-plane](/wiki/Statement:jc-plane.md) — 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 $\mathbb C(p,q)$. [1]
- uses [keller-birational](/wiki/Statement:keller-birational.md) — It then applies Keller's birational theorem, which closes degree 1.
- uses [campbell-galois](/wiki/Statement:campbell-galois.md) — 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.

## References

1. 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).

## Relations to this page

- related to [gallagher-families](/wiki/Statement:gallagher-families.md) — The family has every prime $p\ge3$ as a fiber degree in dimension three, so the prime-degree obstruction claimed for the plane does not extend to ℂ³.

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