## 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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