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We associate an Albert form to any pair of cyclic algebras of prime degree p over a field F with $${\text {char}}(F)=p$$ which coincides with the classical Albert form when $$p=2$$ . We prove that if every Albert form is isotropic, then $$H^4(F)=0$$ . As a result, we obtain that if F is a linked field with $${\text {char}}(F)=2$$ , then its u-invariant is either 0, 2, 4, or 8. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic not 2 and A a central simple algebra with an involution σ. A result of Mahmoudi provides an upper bound for the u-invariants of hermitian forms and skew-hermitian forms over ( A, σ) in terms of the u-invariant of K. In this paper we give a different upper bound when A is a tensor product of quaternion algebras and σ is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi. [ABSTRACT FROM AUTHOR]

We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik's symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik's construction of fields of u-invariant 2 + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2. [ABSTRACT FROM AUTHOR]