Mathematics 5392 Non-associative Algebras
Non-associative algebras: An Overview
- Give an irreducible polynomial of degree 4 over GF(2).
- Construct the field of 16 elements.
Semifields Find two elements in System V that do not commute. Find three elements in System V that do not associate.
Automorphisms of the semifield given by System V.
Review of automorphisms of a field
Show that each nuclei is a field.
Orders of semifields Prove: The order of a proper semifield is greater than or equal to 16.
Finite Affine and Projective Planes
Coordinatization. Coordinatize the Fano plane.
Isotopisms and autotopisms of semifields Thursday 3/4 Primitive elements Compute the matrix representation of System V. Tuesday 3/9 Matrix representations Compute the matrix representation of System W. Thursday 3/11 Wene's paper on the multiplicative structure of division algebras March 15-19 Spring Break Tuesday 3/23 Semifields of order 16 are left and right primitive Find a primitive element in System V
Semifields of order 32 are left and right primitive Find a primitive element in the binary semifield with 32 elements
Semifields three-dimensional over a field are right primitive
Knuth binary semifields
Automorphisms of Knuth binary semifields
Open problems on semifields (Wene's paper) Tuesday 4/13 Knuth Cubical Array Thursday 4/15 Semifield Planes Tuesday 4/20 Linlin's presentation Thursday 4/22 Collineations of Semifield Planes Tuesday 4/27 The Autotopism Group of a Semifield Plane Thursday 4/29 Isotopisms and the Theorem of Albert
Examples of Semifields Thursday 5/6 Examples of Semifields