Does $2y^2=x^4-17$ have a root in every P-adic number field?
My answer is yes, using Hensels lemma.
Let $f(x)=x^4-17$ (all coefficients have P-adic norm less than $1$ for each P-adic norm), let $a$=1, then for $p \neq 2$ we have
$|f(a)|_p=|16|_p=1 < 4=|4a^3|_p^2=|f'(a)|_p^2$
and so $f$ has a root by Hensels lemma.
For $p=2$ we have
$|f(a)|_2=16^{-1}<4^{-1}=|f'(a)|_2^2$
so again, by Hensels lemma, there is a root.
Is this correct? It feels too easy.