  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.
