Integers in p–adically closed fields are definable

We show that the integers in p–adically closed fields are definable. 1 Theory of-adically closed flelds In this short memo we show that the integers in p–adically closed fields are definable. This is a simple generalization of the fact that the integers in $\mathbb{Q}_{\mathrm{p}} $ are definable. First we need to fix a language for the model theory of p–adically closed fields. The language $\mathcal{L}_{R}=\{+,-, \cdot,-1,R,P_{n}(n\in \mathrm{N}),0,1.\pi,u_{1}, \cdots u_{d-1}\} $ , where $R $ and $P_{n} $ are unary predicates, $\pi,u_{1}, $ $\cdots,\mathrm{u}_{d-1} $ are constants. The axiom of p–adically closed fields is the infinite set of following sentences. $\bullet $ $\mathrm{t}\mathrm{h}\infty \mathrm{r}\mathrm{y} $ of fields of characterestic zero $\bullet $ $\forall x(x\neq 0arrow R(x)\vee R(x^{-1}))$ $\bullet $ $\forall x(P_{n}(x)rightarrow\exists y(y^{n}=x)) $ for each $n $. $\bullet $ $\pi $ is a prime element: this means that $v(\pi) $ is the least positive element, i.e., $v(\pi)>$ $\mathrm{O}\wedge\forall x(v(x)\geq 0arrow v(\pi)<v(x\rangle) $ which can be expressed by $R(\pi)\wedge\neg R(\pi^{-1})\wedge\forall x(R(x)arrow$ $R(x\pi^{-1}) $ A $\neg R(\pi x^{-1})) $ (for the definition of a prime element, see p. 13 of [1]