Division algebras over $C_{2}$- and $C_{3}$-fields

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, ...

Triangulaizability of Algebras Over Division Rings

Kummer subfields of tame division algebras over Henselian valued fields

By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p over E such that p 6= char(Ē) and rk(ΓD/ΓF ) ≥ 3 [re...

triangulaizability of algebras over division rings

Fields of Definition for Division Algebras

Let A be a finite-dimensional division algebra containing a base field k in its center F . We say that A is defined over a subfield F0 if there exists an F0-algebra A0 such that A = A0⊗F0 F . We show that: (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n ≥ 5, then A is defined over a field F0 of transcendence degree ≤ 1 2 (n − 1)(n − 2) over k. (iii)...