The Witt ring kernel for a fourth degree field extension and related problems

Abstract

a b s t r a c tIn the first part of this paper we compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n-fold Pfister forms.In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras.In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch-Kato conjecture for exponent 2, which was proven by Voevodsky.

Witt kernels and Brauer kernels for quartic extensions in characteristic two

“…The complete determination of Witt kernels for arbitrary degree 4 extensions is due to Sivatski [22]. To formulate his result, first note that in characteristic not 2 the degree 4 extension E/F will be separable and hence simple, say, E = F (α).…”

Cohomological operations for the Brauer group and Witt kernels of a quartic field extension

“…Indeed, conditions (3) and (4) are obvious. Condition (2) follows from ( [9], Corollary 4). Condition (1) is clear if L is not an etale F -algebra, for in this case the form ϕ is degenerate, hence isotropic.…”

“…Obviously, (4) follows from parts (1)−3). By ( [9], Corollary 2) τ = m i=1 π i , where each π i is similar to a two-fold or a one-fold Pfister form, and π iL = 0 for every i. Note that w n (π i ) = 0 for n ≥ 3, w 2 (π i ) = π i if π i is two-fold, and w 2 ( t, tu ) = t, u , for any t, u ∈ F * .…”

“…The proof of Theorem 9 below is based on the result on γ 2 ( 2 Br(L/F )) obtained in Corollary 3 and the description of W (L/F ) in [9].…”

Tores associés à une algèbre étale quartique

“…S/C est stablement rationnel. Considérons ensuite la suite exacte (18). Comme H 1 (k(T 1 S/C ), G m ) = 1, le même argument que dans la première partie de la preuve montre que U est birationnellement équivalent à T 1 S/C × G m , donc U est stablement rationnel.…”