Does $2y^2=4+17x^4$ have solutions in $\Bbb{Q}_2(\sqrt{-3})$ ? My try is the following, is this correct ?
Look at $2$ adic valuation $v$ of $\Bbb{Q}_2(\sqrt{-3})$ on both side.
$v(2y^2)=1+2v(y)$,
$v(4+17x^4)=min\{2,4v(x)\}$(because $2=v(4)\neq v(17x^4)=4v(x)$).
But $1+2v(y)$ and $min\{2,4v(x)\}$ never equals because $v$ is discrete valuation.
Thus $2y^2=4+17x^4$ does not have solution in $\Bbb{Q}_2(\sqrt{-3})$$■$
Thank you for your help.