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The elementary theory of free pseudo p-adically closed fields of finite corank

Published online by Cambridge University Press:  12 March 2014

Ido Efrat
Ido Efrat*
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
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Extract

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):

(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.

(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.

(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.

In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 484 - 496
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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