Kummer-faithfulness of k(σ) for e = 1

Ascertain whether, for a finitely generated field k of characteristic zero and σ ∈ G_k, the field k(σ) (the fixed field of σ in an algebraic closure of k) is Kummer-faithful.

Background

Kummer-faithfulness means that for every finite extension L of the base field and every semiabelian variety A over L, the divisible part A(L)div of the Mordell–Weil group vanishes. The paper proves Kummer-faithfulness for many large extensions (e.g., K[σ] for e ≥ 1 almost surely, and various fields generated by torsion or division points), but explicitly notes that the status for k(σ) (with e = 1) is unknown.