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.
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