Roberto Aravire

The behavior of quadratic and differential forms under function field extensions in characteristic two

Abstract Let F be a field of characteristic 2. Let Ω n F be the F-space of absolute differential forms over F. There is a homomorphism ℘ :Ω n F →Ω n F / d Ω n−1 F given by ℘(x d x 1 /x 1 ∧⋯∧ d x n /x n )=(x 2 −x) d x 1 /x 1 ∧⋯∧ d x n /x n mod d Ω F n−1 . Let H n+1 (F)= Coker (℘) . We study the behavior of Hn+1(F) under the function field F(φ)/F, where φ=〈〈b1,…,bn〉〉 is an n-fold Pfister form and F(φ) is the function field of the quadric φ=0 over F. We show that ker (H n+1 (F)→H n+1 (F(φ)))= F· d b 1 /b 1 ∧⋯∧ d b n /b n . Using Katos isomorphism of Hn+1(F) with the quotient InWq(F)/In+1Wq(F), where Wq(F) is the Witt group of quadratic forms over F and I⊂W(F) is the maximal ideal of even-dimensional bilinear forms over F, we deduce from the above result the analogue in characteristic 2 of Knebuschs degree conjecture, i.e. InWq(F) is the set of all classes q with deg(q)⩾n.

Relative Brauer groups in characteristic

This paper gives a description of the relative Brauer group Br(E/F) when F has characteristic p, [E : F] = p, and the Galois group Gal(E 1 /F) is solvable, where E 1 is the Galois closure of E over F.

A note on generic splitting of quadratic forms

Let Fbe a field of any characteristic. For n≥ 0, let .The degree conjecture asserts that for each n≥0 Let pbe any n-fold quadratic Pfister form over Fand F(p) the function field of p. Then the function field conjecture asserts We prove that (DC) is equivalent to (FFC).