The fundamental form of an inseparable extension

Abstract:

If K is a finite purely inseparable extension of a field k, then the symmetric multiderivations of K (symmetric maps $f:K \times \cdots \times K\;(n\;{\text {times}}) \to K$ which are derivations as functions of each single variable) form a ring under the symmetrized cup product. This ring contains an element $\Gamma (K/k)$ called the fundamental form of K over k, which is defined up to multiplication by a nonzero element of K and has the property that if B is any intermediate field between K and k, then $\Gamma (K/B)$ divides $\Gamma (K/k)$.