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Commutative presemifields and semifields. (English) Zbl 1194.12007

Summary: Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely, any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert’s commutative twisted fields. A classification of all planar Dembowski-Ostrom polynomials over any finite field of order \(p^3\), \(p\) an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials \(X^{10}+aX^6 - a^2X^2\) with \(a\neq 0\) describes precisely two new commutative presemifields of order \(3^e\) for each odd \(e\geq \)5.

MSC:

12K10 Semifields
12E20 Finite fields (field-theoretic aspects)
17A35 Nonassociative division algebras
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry

Software:

Magma
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Full Text: DOI

References:

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