I want to show
$p$-adic (normed, local, henselian ring) rings is definable in $p$-adic field by a first order formula.
If $p\neq 2$ we have $$\forall x\in \mathbb{Q}_{p}, \,\,\, x \in \mathbb{Z}_{p} \leftrightarrow \exists y \,\,\, y^{2}=1+px^{2}.$$ To prove this we use the henselian property of $p$-adic rings and this fact $$\mathbb{Z}_{p}=\{x\in \mathbb{Q}_{p}\,\, \vert \,\, |x|\leq 1\}.$$ More precisely,
If $x\in\mathbb{Z}_{p}$ then $\overline{f}(1)=0$ and $\overline{f}(1)\neq 0$ by henselian exists $y$ is true in equality. If $x$ be true in equality then $|x|\leq 1$ by use valuation map $$v(x)=n \leftrightarrow p^{n}\mid x,\,\, p^{n+1}\nmid x$$ and $|x|=2^{-v(x)}$.
But if $p=2$ we have $$\forall x\in \mathbb{Q}_{2}, \,\,\, x \in \mathbb{Z}_{2} \leftrightarrow \exists y \,\,\, y^{2}=1+8x^{2}$$ I can't prove this because if $x\in\mathbb{Z}_{2}$ then $\overline{f}(1)=0$ and $\overline{f}(1)=0$ in $\mathbb F_{2}$.
Please help me.