UTA Department of Mathematics

Metroplex Algebraic Geometry, Algebra and Number Theory (AGANT) Seminar,
a joint venture of   UNT,   UTA   and   TCU

Date/Time/Room: Friday (11/14/2008) at 4:00 pm in 304 Pickard Hall

Speaker: Dr. Vikram Jha, Professor,
Glasgow Caledonian University, Scotland.

"The Geometry of Finite-Fractional-Dimensional Division Rings"

Abstract: Wedderburn's theorem states that every finite associative division ring is a Galois field. L. E. Dickson proved the converse to be false by constructing an infinite class of finite strictly non-associative division rings (or `semifields' as they are now usually called) that are commutative. Since then, many classes of semifields have been discovered. In particular, the semifields of Albert and Knuth together imply complete knowledge of the orders of the finite semifields: a (strictly non-associative) semifield of order n exists iff n=pr, where r>2 and p is a prime number.
Galois fields and semifields have many intriguing similarities and differences. This talk will focus on an important difference associated with the dimensions of a field and its subfields, over a selected `base' field such as the prime field. Thus, any subfield of the Galois field F=GF(qn), with chosen base GF(q), has form K=GF(qm), and we regard n/m as one of the `dimensions' of F. It is a well-known fact that the dimensions of any finite field are always integers. Until very recently, it was widely believed -- indeed, even believed to be proven -- that this was also true for semifields: if K is a sub-semifield of a semifield D, then |D|=|K|n for some integer n. However, this belief has been refuted by the recent emergence of infinitely many semifields of even order and fractional dimension.
The aim of the talk is to present a powerful technique for detecting when the plane associated with a given semifield, may be recoordinatized with a (possibly different) semifield that is guaranteed to admit a fractional dimension. The method is not restricted to the characteristic-2 case, and, in addition to detecting semifields D that have fractional dimension, it also finds sub-semifields E inside D such that n= log (to the base |E|) of |D| is a (non-integer) fraction.
The work described is part of recent joint work with Professor Minerva Cordero.

A map of UTA may be found at http://www.uta.edu/maps/, and directions to UTA are on that website.
Note that PKH is the triangular building in the central-campus region of the map. Select Pickard Hall from the building selector. PKH is next to the ``garage'' & is on Nedderman Dr between Planatarium Place (aka College St) & West St. West St is one-way northbound between W Mitchell St and 3rd St. Parking is available in the nearby (East and SE of PKH) parking lots with a UTA permit (but many people park there late on Fridays with no permit and are not caught). There are a few metered parking spots on the East side of Pickard Hall on West St. Visitor (free), & metered, parking is available by Davis Hall (marked DH in SW part of the submap). A parking garage (expensive) is nextdoor to PKH (use West St to get to it). Note that if heading south on Cooper, then no left turn is allowed to Nedderman Dr.
Note that PKH 304 is on the 3rd floor of PKH; turn right out of elevator on 3rd floor.