Finite Semifields are division algebras in which multiplication is not assumed to be associative. They have been studied in many contexts over the years; as algebraic objects, as the coordinatisation of projective planes with certain symmetries, and more recently as rank-metric codes. Determining the equivalence of two semifields is difficult, and so various equivalence invariants have been proposed and studied.
In this talk we will discuss some of these invariants, with particular emphasis on recent work with Michel Lavrauw on the tensor rank. The tensor rank is an invariant naturally arising from multilinear algebra, and can be viewed as a measure of multiplicative complexity. We present the first known examples of finite semifields of lower tensor rank than the finite field of the same size.