Mathematics 5392  Non-associative Algebras

Spring 2010





Thursday 1/21

Non-associative algebras: An Overview


Tuesday 1/26

Finite Fields
  1. Give an irreducible polynomial of degree 4 over GF(2).
  2. Construct the field of 16 elements.

Thursday 1/28

Semifields Find two elements in System V that do not commute. Find three elements in System V that do not associate.

Tuesday 2/2

Automorphisms of the  semifield given by System V.
Review of automorphisms of a field

Thursday 2/4

Show that each nuclei is a field.

Tuesday 2/9

Knuth's paper  

Thursday 2/11

Knuth's paper  

Tuesday 2/16

Orders of semifields Prove: The order of a proper semifield is greater than or equal to 16.

Thursday 2/18

Finite Affine and Projective Planes  

Tuesday 2/23

Coordinatization. Coordinatize the Fano plane.

Thursday 2/25

Ternary Rings  

Tuesday 3/2

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

Thursday 3/25

Semifields of order 32 are left and right primitive Find a primitive element in the binary semifield with 32 elements

Tuesday 3/30

Semifields three-dimensional over a field are right primitive  

Thursday 4/1

Knuth binary semifields  

Tuesday 4/6

Automorphisms of Knuth binary semifields  

Thursday 4/8

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  

Tuesday 5/4

Examples of Semifields  
Thursday 5/6 Examples of Semifields  

Tuesday 5/11

Final Exam