Does $ \Bbb{Q}_2$ has $\sqrt{-1}$? Does $ \Bbb{Q}_2$ has $\sqrt{-1}$?
I tried to use Hensel lemma as usual.
Let $f(x)＝x^2＋1$.
But if some $a∈\Bbb{Z}$, $f(a)＝0$, then $f'(a)$ can always divide by $2$.
So I cannot use Hensel lemma.
Could you help me?

No: if the monic polynomial $x^2+1$ had a root in $\Bbb Q_2$ then that root would be in the ring of integers $\Bbb Z_2$; but a root in $\Bbb Z_2$ implies that there is a root modulo $2^k$ for all $k\ge1$, contradicting the fact that $x^2+1$ has no root modulo $4$.
