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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 $\sqrt{-1}\in F$

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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.

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