We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 3 8 with left nucleus of order 3 and middle nucleus of order 3 2 .

This note summarises a recent search for commutative semifields of order 243 and 3125. For each of these two orders, we use the correspondence between commutative semifields of odd order and planar Dembowski-Ostrom polynomials to classify those commutative semifields which can be represented by a planar DO polynomial with coefficients in the base field. The classification yields a new commutative semifield of each order. Furthermore, the new commutative semifield of order 243 describes a skew Hadamard difference set which is also new.

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