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Rings of \(p\)-adic functions. (Anneaux de fonctions \(p\)-adiques.) (French) Zbl 0868.03017

Summary: We study first-order properties of the quotient rings \({\mathcal C}(V)/{\mathcal P}\) by a prime ideal \(\mathcal P\), where \({\mathcal C}(V)\) is the ring of \(p\)-adic valued continuous definable functions on some affine \(p\)-adic variety \(V\). We show that they are integrally closed Henselian local rings, with a \(p\)-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy \(\forall x,y\) \((x|y^2\vee y|x^2)\).

MSC:

03C60 Model-theoretic algebra
13J07 Analytical algebras and rings
13L05 Applications of logic to commutative algebra
12J10 Valued fields
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References:

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