A note on highly Kummer-faithful fields

We introduce a notion of highly Kummer-faithful fields and study its relationship with the fields. also give some examples For example, if $k$ is number field finite degree over \mathbb{Q}, $g$ an integer $> 0$ $\mathbf{m}=(m_p)_p$ family non-negative integers, where $p$ ranges all prime numbers, then extension $k_{g,\mathbf{m}}$ obtained by adjoining to coordinates elements $p^{m_p}$-torsion subgroup $A[p^{m_p}]$ $A$ for semi-abelian varieties dimension at most numbers $p$, Kummer-faithful.