# Keller's birational theorem

- kind: theorem
- status: proved (evidence tier V3)
- year: 1939
- authors: O.-H. Keller
- in one line: A birational Keller map is an automorphism

**Statement.** Let $\operatorname{char}k=0$ and $F\in k[x_1,\dots,x_n]^n$ with $\det JF\in k^{\times}$. If $k(F_1,\dots,F_n)=k(x_1,\dots,x_n)$, then $k[F]=k[x]$.

Keller proved this in the paper where he posed the conjecture. He worked over the integers with functional determinant 1, and showed that if the map is rationally invertible, its inverse is polynomial with integer coefficients. The proof used homology.[1] In modern terms, the conjecture holds for Keller maps of field degree 1, so [Keller's theorem is the Jacobian conjecture restricted to birational maps](/wiki/Statement:jc-n.md).

The trivial extension $k(x)=k(F)$ is Galois, so [Keller's theorem is the degree-1 case of Campbell's Galois criterion](/wiki/Statement:campbell-galois.md).

It is the first wall around every counterexample. A non-invertible Keller map has field degree at least 2, and in fact at least 3 (see [the degree spectrum](/wiki/Statement:degree-spectrum-c3.md)). Alpöge's map has degree 3.

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — The proof used homology. In modern terms, the conjecture holds for Keller maps of field degree 1, so Keller's theorem is the Jacobian conjecture restricted to birational maps. [1]
- special case of [campbell-galois](/wiki/Statement:campbell-galois.md) — The trivial extension $k(x)=k(F)$ is Galois, so Keller's theorem is the degree-1 case of Campbell's Galois criterion.

## References

1. O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939) 299–306; zbMATH review by Kneser.

## Relations to this page

- used by [degree-spectrum-c3](/wiki/Statement:degree-spectrum-c3.md) — Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem.
- used by [prime-degree-plane](/wiki/Statement:prime-degree-plane.md) — It then applies Keller's birational theorem, which closes degree 1.
- used by [proper-etale-bijective](/wiki/Statement:proper-etale-bijective.md) — (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.

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