Kummer-faithful fields with finitely generated absolute Galois group

This paper studies the structure of the Mordell–Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single element. A probabilistic argument using the Haar measure on the absolute Galois group of a number field shows that almost all such fields are Kummer-faithful, i.e., the Mordell–Weil group of any semiabelian variety over any finite extension of such a field has trivial divisible part. This result implies that there exists a Kummer-faithful field algebraic over a number field whose absolute Galois group is abelian.