Albert's Construction for Semifields of Even Order

Abstract

Albert's construction for commutative semifields of order 2 ⁿ , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism.

The authors present an explicit construction of an isotope of the 2⁵-element semifield that contains a subsemifield of order 2².

Key Words:

• Binary semifields
• Semifields
• Translation planes

2000 Mathematics Subject Classification:

• Primary 51A40, 17A35
• Secondary 05B25, 51A35, 51A50

Notes

Communicated by A. Elduque.