Kummer-faithfulness of k(σ) with σ ∈ G_k

Establish whether, for a number field k and σ in its absolute Galois group G_k, the fixed field k(σ) is Kummer-faithful; that is, whether for every finite extension L of k(σ) and every semiabelian variety A over L the Mordell–Weil group A(L) has trivial divisible part.

Background

The paper proves Kummer-faithfulness for many large extensions, including finite extensions of K[σ] for e≥2 and toral Kummer-faithfulness for finite extensions of K(σ) even when e=1. However, the full Kummer-faithfulness of fields of the form k(σ) (e=1) remains unresolved.