A note on highly Kummer-faithful fields. (English) Zbl 1498.11230
Summary: 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.
MSC:
| 11S15 | Ramification and extension theory |
| 11S20 | Galois theory |
| 14K05 | Algebraic theory of abelian varieties |
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