Differential forms and bilinear forms under field extensions

Let F   be a field of characteristic p>0p>0. Let Ωn(F)Ωn(F) be the F-vector space of n-differentials of F   over FpFp. Let K=F(g)K=F(g) be the function field of an irreducible polynomial g   in m⩾1m⩾1 variables over F  . We derive an explicit description of the kernel of the restriction map Ωn(F)→Ωn(K)Ωn(F)→Ωn(K). As an application in the case p=2p=2, we determine the kernel of the restriction map when passing from the Witt ring (resp. graded Witt ring) of symmetric bilinear forms over F to that over such a function field extension K.