Witt kernels and Brauer kernels for quartic extensions in characteristic two

Abstract

Abstract. Let F be a field of characteristic 2 and let E/F be a field extension of degree 4. We determine the kernel Wq(E/F ) of the restriction map WqF → WqE between the Witt groups of nondegenerate quadratic forms over F and over E, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduct the corresponding result for the Witt kernel W (E/F ) of the restriction map W F → W E between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As application, we describe the 2-torsion part of the Brauer kernel for such extensions.

Cohomology and graded Witt group kernels for extensions of degree four in characteristic two

“…We note that the Witt kernels of all degree four extensions in characteristic 2 have been recently found by Hoffmann and Sobiech [4], so the results in this paper generalize some of these results to the graded case. More details are provided in Remark 2.…”

“…(iii) Hoffmann and Sobiech [4] also determined Brauer kernels for degree four extensions. By the calculation in (ii), our calculation of H 2 2 (E/F ) in the cyclic and dihedral cases (see Theorem 39) coincides with the calculation of Hoffmann-Sobiech for the relative Brauer group.…”

“…In the case of A 4 , after extending the base to the splitting of the resolvent cubic L , we have E/L a biquadratic extension, and in the case of S 4 we have E/L a cyclic degree 4 extension. The Witt and Brauer kernels for the general E/F of degree 4 has been calculated by Hoffmann and Sobiech [4]. We next record the notation that will be in force throughout the rest of the paper, depending upon which of the three cases is being discussed.…”

“…(ii) Hoffmann and Sobiech [4] have recently determined Witt kernels for extensions of degree 4 in characteristic two. Their formulation of generators, although different than those in Definition 2 can be checked to be equivalent.…”

“…For example, in the dihedral case, the irreducible polynomial of δ is readily checked to be x 4 +(1 +s)x 2 +sx +(r 2 +rs +bs 2 ) which has as resolvent cubic R(x) = x 3 + (1 + s)x 2 + s 2 = (x + s)(x 2 + x + s). Hoffmann and Sobiech [4,Theorem 5.4] show that the Witt kernel W q (E/F ) is generated as a W F -module by [1, b] and ψ(f ) := R(f ), (r 2 + rs + bs 2 )f −2 ]] where f ∈ F . We verify that the Hoffmann-Sobiech generators coincide with I 2 r,s,b F in this case.…”