Nicely semiramified division algebras over Henselian fields

We recall that a nicely semiramified division algebra is defined to be a defectless finitedimensional valued central division algebra D over a field E with inertial and totally ramified radical-type (TRRT) maximal subfields [7, Definition, page 149]. Equivalent statements to this definition were given in [7, Theorem 4.4] when the field E is Henselian. These division algebras, as claimed in [7, page 128], appeared in [10] as examples of division algebras with nonzero SK1. The main purpose of this paper is to prove that over a Henselian field E, any central division algebra with inertial and totally ramified maximal subfields (resp., split by inertial and totally ramified field extensions of E) is nicely semiramified. We precise that all rings considered in this work are assumed to be associative with a unit and all free modules are assumed to be finite-dimensional. A valued division algebra D over a field E—we adopt the same valuative definitions as in [7]—is called defectless (over E) if [D : E] = [D̄ : Ē](ΓD : ΓE), where [D̄ : Ē] (resp., (ΓD : ΓE)) is the residue degree (resp., ramification index) of D over E. We recall that for any valued central division algebra D over a field E, the center Z(D̄) of D̄ is a normal field extension of Ē and the mapping