Commutative presemifields and semifields

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 p3, p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X10+aX6−a2X2 with a≠0 describes precisely two new commutative presemifields of order e3 for each odd e⩾5.