*Mathematics 5392
Non-associative Algebras*

*Spring 2010*

Date

Section/Topic

HomeworkThursday 1/21

Non-associative algebras: An Overview

http://www-math.cudenver.edu/~wcherowi/courses/finflds.html

Tuesday 1/26

Finite Fields

- Give an irreducible polynomial of degree 4 over GF(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 fieldThursday 2/4

Pre-semifields.

NucleiShow 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