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Preprint Number 1620
1620. Nathanaël Mariaule Expansions of the p-adic numbers that interprets the ring of
integers E-mail: Submission date: 27 May 2019 Abstract: Let \widetilde ℚ_p be the field of p-adic numbers in the language of rings. In this paper we consider the theory of \widetilde ℚ_p expanded by two predicates interpreted by multiplicative subgroups α^ℤ and β^ℤ where α, β in ℕ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p-adic valuation. If either α or β has zero valuation we show that the theory of (\widetilde ℚ_p, α^ℤ, β^ℤ does not interpret Peano arithmetic. In that case we also prove that the theory is decidable iff the theory of (\widetilde ℚ_p, α^ℤ ˙ β^ℤ) is decidable. Mathematics Subject Classification: 03C65, 03C10 Keywords and phrases: |

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