# Model theory of the field of $p$-adic numbers expanded by a multiplicative subgroup

@article{Mariaule2018ModelTO, title={Model theory of the field of \$p\$-adic numbers expanded by a multiplicative subgroup}, author={Nathanael Mariaule}, journal={arXiv: Logic}, year={2018} }

Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that $G$ satisfies Mann property and is small as subset of a first-order structure. First, we give an axiomatisation of the first-order theory of this structure. This includes an axiomatisation of the theory of the group $G$ as valued group (with the valuation induced on $G$ by the $p$-adic valuation). If the subgroups $G^{[n]}$ of $G$ have finite… Expand

#### 4 Citations

Expansions of the p-adic numbers that interpret the ring of integers

- Mathematics, Computer Science
- Math. Log. Q.
- 2020

The theory of this structure expanded by two predicates interpreted by multiplicative subgroups interprets Peano arithmetic if $\alpha$ and $\beta$ have positive $p$-adic valuation and if either $alpha$ or $beta$ has zero valuation is shown. Expand

Nippy proofs of p-adic results of Delon and Yao

- Mathematics
- 2020

Let $K$ be an elementary extension of $\mathbb{Q}_p$, $V$ be the set of finite $a \in K$, $\mathrm{st}$ be the standard part map $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ be $K$-definable.… Expand

Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang

- Mathematics
- 2020

This is a contribution to the classification problem for dp-minimal expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on $(\mathbb{Z},+)$. We use results on "dense pairs" to… Expand

Externally definable quotients and NIP expansions of the real ordered additive group

- Mathematics
- 2019

Let $\mathscr{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R}, 0$ and collection $\mathcal{B}$ of bounded subsets of $\mathbb{R}^n$ such that $(\mathbb{R},<,+,\mathcal{B})$ is o-minimal.

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