# Campbell's theorem: Galois Keller maps are automorphisms

- kind: theorem
- status: proved (evidence tier V3)
- year: 1973
- authors: L. A. Campbell (over ℂ, 1973); M. Razar (1979) and D. Wright (1981) over any field of characteristic 0
- in one line: If ℂ(x)/ℂ(F) is Galois, a Keller map F is an automorphism

**Statement.** Let $F:\mathbb C^n\to\mathbb C^n$ be polynomial with $\det JF\in\mathbb C^{\times}$. If the field extension $\mathbb C(F)\subset\mathbb C(x)$ is Galois, then $F$ is a polynomial automorphism.

For a Keller map, the degree of $\mathbb C(x)/\mathbb C(F)$ is the number of preimages of a general point. Campbell's theorem says that if this extension is Galois, the map is invertible, so [the Jacobian conjecture holds for every Keller map whose function field extension is Galois](/wiki/Statement:jc-n.md).[1]

Every extension of degree 2 in characteristic zero is Galois. So no counterexample, in any dimension, has geometric degree 2, and the least possible degree is 3. Alpöge's map attains it, and its cubic extension is not Galois: its Galois closure has group $S_3$. The research for this page deduced that from the theorem and the fiber count; it is not quoted from a source.

In the plane, more is known. Degree 3 was excluded by Orevkov (1986) and degree 4 by Domrina, and Żołądek proved the conjecture for every topological degree up to 5. A preprint claims every prime degree is excluded (see [prime degree in the plane](/wiki/Statement:prime-degree-plane.md)).

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — Campbell's theorem says that if this extension is Galois, the map is invertible, so the Jacobian conjecture holds for every Keller map whose function field extension is Galois. [1]

## References

1. L. A. Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973) 243–248, [doi:10.1007/BF01349234](https://doi.org/10.1007/BF01349234). M. Razar, Polynomial maps with constant Jacobian, Israel J. Math. 32 (1979) 97–106. D. Wright, On the Jacobian conjecture, Illinois J. Math. 25 (1981) 423–440.

## Relations to this page

- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — So Alpöge's map attains the least degree that Keller's and Campbell's theorems leave open, and its cubic extension is not Galois. This observation, with the resulting minimality of 3, was recorded in the Fong–Fable ledger on 20 July.
- used by [degree-spectrum-c3](/wiki/Statement:degree-spectrum-c3.md) — For degree 2, a quadratic extension in characteristic zero is Galois, and Campbell's theorem makes every Galois Keller map an automorphism.
- generalizes [keller-birational](/wiki/Statement:keller-birational.md) — The trivial extension $k(x)=k(F)$ is Galois, so Keller's theorem is the degree-1 case of Campbell's Galois criterion.
- used by [prime-degree-plane](/wiki/Statement:prime-degree-plane.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.

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