Results on Witt kernels of quadratic forms for multi-quadratic extensions

Abstract

In this paper we compute the Witt kernel of quadratic forms for the composition of a purely inseparable multi-quadratic extension with a separable quadratic extension. We also include the case of a multi-quadratic purely inseparable extension by completing the proof given before by the second author for such an extension.

Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

“…Corollary 4.6 together with Corollary 4.3 (in the case n = 1, see Remark 4.2) immediately imply the result by Aravire and Laghribi [1] mentioned in the introduction.…”

“…Let K be a finite multiquadratic extension of a field F of separability degree at most 2, in other words, K = F ( √ a 1 , · · · , √ a n ), or K = F ( √ a 1 , · · · , √ a n , ℘ −1 (b)), a i , b ∈ F * , where ℘ −1 (b) is a root of X 2 + X + b. In [1], Aravire and Laghribi computed the kernel W q (K/F ) of the natural map (induced by scalar extension) W q (F ) → W q (K) between the Witt groups of nonsingular quadratic forms over F and K, respectively. They show that…”

“…Actually, they prove more, namely they determine the kernel of the restriction map in Kato's cohomology H n 2 (F ) → H n 2 (K) and then deduce the result on the Witt kernels by using Kato's theorem [7] that H * 2 (F ) is naturally isomorphic to the graded Witt module of quadratic forms over F . For more details, we refer the reader to Aravire and Laghribi's article [1] and the references there.…”

“…In [1], Aravire and Laghribi computed the kernel W q (K/F ) of the natural map (induced by scalar extension) W q (F ) → W q (K) between the Witt groups of nonsingular quadratic forms over F and K, respectively. They show that W q (K/F ) = n i=1 1, a i b ⊗ W q (F ) , in the purely inseparable case, and…”

The $u^n$-invariant and the symbol length of $H_2^n(F)$

“…, u n (F). The main tool we use is Aravire and Laghribi's description of the Witt kernel of a multiquadratic purely inseparable field extension [1,Proposition 2]. This method only works in characteristic 2 because of the existence of purely inseparable quadratic splitting fields.…”