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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
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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
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