I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are equivalent $$(i) \text{ There exists } b \in \mathbb{Q}_{p} \text{ such that } u = b^2.$$ $$(ii) \text{ There exists }b \in \mathbb{Z}_{p}^{\times} \text{ such that }u = b^2.$$ $$(iii) \text{ There exists }\bar{b} \in \mathbb{F}_{p} \text{ s.th. if } \bar{u} \text{ denotes the reduction of } u \text{ modulo the ideal } p\mathbb{Z}_{p}, \text{ then }\bar{u} = \bar{b}^2.$$
(2) Using the above remark, prove that there exists a unit $u_0 \in \mathbb{Z}_{p}^{x}$ such that, for all $b \in \mathbb{Q}_p, \; u_0 \neq b^2$.
(3) Let $u_0$ be as in (2), let $a \in \mathbb{Q}_p$, and let $K = \mathbb{Q}_{p}(\sqrt{a})$.
Prove that $ K \in \lbrace \mathbb{Q}_p, \mathbb{Q}_{p}(\sqrt{u_{0}}), \mathbb{Q}_{p} (\sqrt{p}),\mathbb{Q}_p (\sqrt{u_{0} p})\rbrace$.