Witt kernels of bilinear forms for algebraic extensions in characteristic 2

Abstract

Let F F be a field of characteristic 2 2 and let K / F K/F be a purely inseparable extension of exponent 1 1 . We determine the kernel W ( K / F ) W(K/F) of the natural restriction map W F → W K WF\to WK between the Witt rings of bilinear forms of F F and K K , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension K / F K/F . Based on this result, we will determine W ( K / F ) W(K/F) for a wide class of finite extensions which are not necessarily purely inseparable.