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- W2009139840 abstract "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 σ 1 ,…,σ e + m ϵ G(K) (in the sense of the unique Haar measure on G(K) e+m ): (a) K σ is pseudo p -adically closed (abbreviation: P p C), i.e., each nonempty absolutely irreducible variety defined over K σ has a K σ -rational point, provided that it has a simple rational point in each p -adic closure of K σ . (b) G ( K σ ) ≅ D e,m , where D e,m is the free profinite product of e copies Γ 1 ,…, Γ e of G (ℚ p ) and a free profinite group of rank m . (c) K σ has exactly e nonequivalent p -adic valuation rings. They are the restrictions O σ1 ,…, O σ e 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)." @default.
- W2009139840 created "2016-06-24" @default.
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- W2009139840 date "1991-06-01" @default.
- W2009139840 modified "2023-10-16" @default.
- W2009139840 title "The elementary theory of free pseudo p-adically closed fields of finite corank " @default.
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- W2009139840 doi "https://doi.org/10.2307/2274695" @default.
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