A note on highly Kummer-faithful fields

Abstract

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 , g is an integer > 0 and m = (m_p)_p is a family of non-negative integers, where p ranges over all prime numbers, then the extension field k_g,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.