## Remarks on existentially closed fields and diophantine equations.(English)Zbl 0545.12012

Generalizing the results of N. Greenleaf [Am. Math. Mon. 76, 808–809 (1969; Zbl 0191.049)] and M. B. Nathanson [ibid. 81, 371–373 (1974; Zbl 0282.10008)] the author proves that if the equation $$g^ n=P(f),$$ where $$P$$ is a separable polynomial of degree $$m$$ over a field $$K$$ with characteristic not dividing $$n$$, and $$n$$ and $$m$$ are greater than 2, has a solution $$f, g$$ in a purely transcendental extension $$K(t)$$ of $$K$$, then $$f$$ and $$g$$ necessarily belong to $$K$$.
He also proves that a non-Henselian valued field $$K$$ with algebraically closed residue field is not existentially closed in any Henselian valued extension field $$L$$, and that $$L/K$$ is not a purely transcendental extension.

### MSC:

 12E12 Equations in general fields 11D41 Higher degree equations; Fermat’s equation 12J10 Valued fields 12L12 Model theory of fields 12F20 Transcendental field extensions

### Citations:

Zbl 0191.049; Zbl 0282.10008
Full Text:

### References:

 [1] L. Van Den Dries , Model Theory of Fields, Decidability and Bounds for Polynomial Ideals , Thesis, University of Utrecht , 1978 . [2] N. Greenleaf , On Fermat’s equation in C(t) , Amer. Math. Monthly , 76 ( 1969 ), pp. 808 - 809 . MR 250973 | Zbl 0191.04905 · Zbl 0191.04905 [3] R. Liouville , Sur l’impossibilité de la relation algébriqne Xn + Yn + Zn = 0 , C. R. Acad. Sci. Paris , 87 ( 1879 ), pp. 1108 - 1110 . JFM 11.0138.03 · JFM 11.0138.03 [4] M. Natanson , Catalan’s equation in k(t) , Amer. Math. Monthly , 81 ( 1974 ), pp. 371 - 373 . MR 335436 | Zbl 0282.10008 · Zbl 0282.10008 [5] P. Ribenboim , Théorie des Valuation , Presses Université de Montréal , 1964 . Zbl 0139.26201 · Zbl 0139.26201 [6] P. Ribenboim , On the completion of a valuation ring , Math. Annalen , 155 ( 1964 ), pp. 392 - 396 . MR 164960 | Zbl 0136.32102 · Zbl 0136.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.