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A note on highly Kummer-faithful fields

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