Every finitely generated regular field extension has a stable transcendence base |
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Authors: | Konrad Neumann |
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Affiliation: | (1) Mathematical Institute, University of Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Germany |
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Abstract: | A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex 1, …,x n with the following properties: The field extensionF/K(x 1,…,x n ) is separable and the Galois hull ofF/K(x 1,…,x n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point. |
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