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An isomorphism theorem for Henselian algebraic extensions of valued fields. (English) Zbl 0773.12004

The paper is motivated by the following problem. Given a valued field \(\mathbb{K}=(K,v)\), find elementary invariants to classify, up to isomorphism over \(\mathbb{K}\), the algebraic Henselian valued field extensions of \(\mathbb{K}\). A partial solution of the problem is given in [A. Prestel and P. Roquette, Formally \(p\)-adic fields (Lect. Notes Math. 1050) (1984; Zbl 0523.12016), Corollary 3.11].
The main goal of the paper is to suitably extend this result to a more general case including that of valued fields of characteristic zero. Let \(\mathbb{K}=(K,v)\) be a valued field. Denote by \(O_{\mathbb{K}}\) its valuation ring, by \(vK\) the value group. Let \(vK^{\geq 0}=\{\delta\in vK\mid\delta\geq 0\}\). For \(\delta\in vK^{\geq 0}\), let \(\mu_ K^ \delta\) be the ideal \(\{a\in O_ K\mid va>\delta\}\) of \(O_ K\). Denote by \(O_ K^ \delta\) the factor ring \(O_ K/\mu_ K^ \delta\). Consider the multiplicative groups \(G_ K^ \delta=K^ \times/1+\mu_ K^ \delta\) for \(\delta\in vK^{\geq 0}\). For \(\delta\in vK^{\geq 0}\) the map \(\theta_ \delta\) of the subset \(\{x\in O_ K^{2\delta}\mid x^ 2\neq 0\}\) into \(G_ K^ \delta\) is defined by \(\theta_ \delta(a+\mu_ K^{2\delta})=a(1+\mu_ K^ \delta)\) for all \(a\in O_ K\) subject to \(va\leq 0\). Consider the system \(\mathbb{K}_ \delta=(O_ K^{2\delta},G_ K^ \delta,\theta_ \delta)\), and call it the mixed \(\delta\)-structure assigned to \(\mathbb{K}\). Now, let \(\mathbb{K}\) be a valued field, \(\mathbb{L}\) an algebraic extension of \(\mathbb{K}\) and \(\mathbb{F}\) a Henselian extension of \(\mathbb{K}\). Then conditions are formulated under which \(\mathbb{L}\) is \(\mathbb{K}\)-embeddable into \(\mathbb{F}\). Similarly, let \(\mathbb{L}\) and \(\mathbb{F}\) be two Henselian algebraic extensions of a valued field \(\mathbb{K}\). Conditions are formulated under which \(\mathbb{L}\) and \(\mathbb{F}\) are \(\mathbb{K}\)-isomorphic.
Reviewer: G.Pestov (Tomsk)

MSC:

12J12 Formally \(p\)-adic fields
12F05 Algebraic field extensions

Citations:

Zbl 0523.12016
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References:

[1] Basarab, S. A.: Relative elimination of quantifiers for Henselian valued fields. Annals of Pure and Applied Logic53 (1991), 51–74 · Zbl 0734.03021 · doi:10.1016/0168-0072(91)90058-T
[2] Kuhlmann, F.-V.: Quantifier elimination for henselian fields relative to additive and multiplicative congruences. In preparation · Zbl 0809.03028
[3] Prestel, A.–Roquette, P.: Formallyp-adic Fields. Lecture Notes Math.1050: Springer (1984) · Zbl 0523.12016
[4] Ribenboim, P.: Théorie des valuations. Les Presses de l’Université de Montréal: Montréal (1968)
[5] Zariski, O.–Samuel, P.: Commutative Algebra, Vol. II: Springer (1960) · Zbl 0121.27801
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