In this question, Is the algebraic closure of a $p$-adic field complete
GEdgar's answer,
Any $x$ which is algebraic of degree $n$ over $\mathbb Q_2$ has a unique series expansion $$ x = \sum 2^{u_j} $$ where $u_j \to \infty$ (unless it is a finite sum) and all $u_j$ are rationals with denominator that divides $n!$
But why this claim holds?
I asked in the comment form, but this answer was posted $9$ years ago, so it is difficult to get reaction.
Thank you in advance.
This seems completely false, even if you make the restriction that @TorstenSchoeneberg suggested in a comment.
For instance, I don’t believe that $\sqrt6$, generator of the totally ramified quadratic extension $\Bbb Q_2(\sqrt6\,)$, can be written in the form quoted by @GEdgar. If I’m wrong here, then I beg Gerry to show us the appropriate expansion.
Most serious of all is the observation of @user8268 that all Edgar’s numbers seem clearly to live only in ramified extensions of $\Bbb Q_2$, while the roots of unity of odd order are not there at all.