Totally Ramified Valuations on Finite-Dimensional Division Algebras

Division algebras $D$ with valuation $v$ are studied, where $D$ is finite-dimensional and totally ramified over its center $F$ (i.e., the ramification index of $v$ over $v{|_F}$ equals $[D:F]$). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When $v{|_F}$ is Henselian a full classification of the $F$-subalgebras of $D$ is given. When $F$ has a Henselian valuation $v$ with separably closed residue field and $A$ is any tame central simple $F$-algebra, an algorithm is given for computing the underlying division algebra of $A$ from a suitable subgroup of ${A^{\ast }}/{F^{\ast }}$. Some examples are constructed using this valuation theory, including the first example of finite-dimensional $F$-central division algebras ${D_1}$ and ${D_2}$ with ${D_1}{ \otimes _F}{D_2}$ not a division ring, but ${D_1}$ and ${D_2}$ having no common subfield $K \supsetneqq F$.