  We construct and describe the basic properties of a family of semifields in
characteristic $2.$ The construction relies on the properties of projective
polynomials over finite fields. We start by associating non-associative
products to each such polynomial. The resulting presemifields form the
degenerate case of our family. They are isotopic to the Knuth semifields which
are quadratic over left and right nucleus. The non-degenerate members of our
family display a very different behaviour. Their left and right nucleus agrees
with the center, the middle nucleus is quadratic over the center. None of those
semifields is isotopic or Knuth equivalent to a commutative semifield. As a
by-product we obtain the complete taxonomy of the characteristic $2$ semifields
which are quadratic over the middle nucleus, bi-quadratic over the left and
right nucleus and not isotopic to twisted fields. This includes {determining}
when two such semifields are isotopic and the order of the autotopism group.
