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Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...

Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...