Perfect nonlinear binomials and their semifields

It is proven that for an appropriate choice of an integer s and α∈GF(p3k) the binomial xps+1−αxpk+p2k+s defines a perfect nonlinear mapping in GF(p3k), which is not equivalent to a monomial one. As a consequence, commutative proper semifields of order p3k are constructed. In most of the cases those are not isotopic to Albert's twisted fields, which are the only previously known examples of such semifields for p⩾5 and odd k>1.