Differential forms and bilinear forms under field extensions (Q493769)

This is an interesting paper. The valuable sections 1--5 collect various known results about ``forms'' over fields \(F\) of finite characteristic \(p>0\) which are needed later. Section \S6 develops a new method for studying field extensions \(F(f)/F\) for an irreducible polynomial \(f(X)= f(X_1,\dots,X_n)\) over \(F\). Surprisingly several results about restriction maps depend largely on the finitely many nonzero coefficients of \(f\). The main results are in Sections 7--9: In \S7 there is an explicit computation of the kernel \(\Omega^m(E/F)\) of \(m\)-differential forms \(\Omega^m(F)\) under the natural map \(F\to E\) for a simple algebraic field extension \(E/F\) (separable or not). Section \S8 contains the corresponding result on \(\Omega^m(E/F)\) for the case \(E=F(f)\) from \S6 which concerns function fields of (affine or projective) hypersurfaces. It is related to the ``norm field'' and the ``norm degree'' of \(f\) over the subfield \(F^p\) of \(p\)-powers in \(F\). Section \S9 is on the Witt ring of symmetric bilinear forms over \(F\) (in the case \(p=2\)) and the function field \(E=F(f)\). The relevant kernels are determined for the 3 cases \(W(E/F)\), \(I^n(E/F)\), and \(\overline I^n(E/F)\), where \(I\) is the fundamental ideal of \(W\) and \(\overline I^n= I^n/I^{n+1}\).