**Simeon Ball**

Universitat Politècnica de Catalunya

A *finite semifield* is a finite set with two operations, addition
and multiplication, such that satisfies all the axioms of a
field except (possibly) associativity of multiplication.

A semifield can be used to coordinatise a projective plane of order
and we are interested in finding semifields that produce non-isomorphic
projective planes. Two semifields are said to be *isotopic* if they
coordinatise isomorphic planes. There are less than roughly 20 known
families of (mutually non-isotopic) semifields. Unless it is immediate that
two semifields are not isotopic it is generally difficult to establish
whether or not they are. Above all, the goal in this area is to construct
many more families of non-isotopic semifields. The first semifields were
discovered by Dickson, roughly 100 years ago with more examples given later
by Albert (1950's), Knuth (1960's), Cohen and Ganley (1980's) and various
families due to Kantor, amongst others, have been constructed in the last
twenty years.

If is finite then it can be shown that for some prime power
and that can viewed as a vector space of rank over
, where multiplication is given by
by the
rule

where is a basis for over . Knuth was first to note that any permutation of the subscripts produces another semifield, so there are six semifields associated with any semifield.

In this talk I shall present a new way to construct finite semifields of order from two subspaces of a vector space of rank over , for some . In fact any finite semifield can be constructed in this way for some , moreover any known semifield of order can be constructed from two subspaces of a vector space of rank or rank over .

The construction also provides us with a new operation (not one of the six due to Knuth) which produces more semifields in the case when .

2005-05-23