Fields of definition for division algebras

Simple item page
dc.creatorLorenz, Martin
dc.creatorReichstein, Zinovy
dc.creatorRowen, Louis H.
dc.creatorSaltman, David J.
dc.date2001-10-18
dc.date2003-06-24
dc.date.accessioned2026-07-07T04:43:55Z
dc.date.available2026-07-07T04:43:55Z
dc.descriptionLet $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In many cases $A$ can be defined over a rational extension of $k$. (2) If $A$ has odd degree $n \ge 5$, then $A$ is defined over a field $F_0$ of transcendence degree at most $(n-1)(n-2)/2$ over $k$. (3) If $A$ is a $Z/m \times Z/2$-crossed product for some $m \ge 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, $M_m(A)$ can be defined over a field $F_0$ of transcendence degree at most 4 over $k$. (4) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree at most 4. (In (1), (3), and (4) we assume that the center of $A$ contains certain roots of unity.)
dc.description24 pages, AMS-LaTeX with xypic. This is the final version of the article, to appear in J. London Math. Soc. More details have been added and several mistakes have been corrected. In particular, the old Theorem 9.1 and Corollary 10.1 have been removed
dc.identifierhttps://arxiv.org/abs/math/0110198
dc.identifierhttp://arxiv.org/abs/math/0110198
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/62432
dc.subjectRings and Algebras
dc.subject16K20;16W22;20C10;11E81
dc.titleFields of definition for division algebras
dc.typetext

Files

Collections

arXiv