Let $F$ be a finite extension of $\mathbb{Q}\_2$, and let $(-,-)\_F$ be the quadratic Hilbert symbol over $F$. Is the following true?


$(-1,-1)\_F=1$ if and only if $\sqrt{-1}\in F$



Hilbert symbol over 2-adic field
No. For example, $(-1,-1)\_F = 1$ for *all* quadratic extensions $F/\mathbf Q\_2$. This was discussed a few days ago on math.stackexchange [here](https://math.stackexchange.com/questions/3965273/over-which-extensions-of-mathbb-q-2-is-x2y2z2-isotropic).


