### A class of symmetric biadditive functionals.

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Let $N/K$ be a Galois extension with Galois group $\mathcal{G}$. We study the set $\mathcal{T}\left(\mathcal{G}\right)$ of $\mathbb{Z}$-linear combinations of characters in the Burnside ring $\mathcal{B}\left(\mathcal{G}\right)$ which give rise to $\mathbb{Z}$-linear combinations of trace forms of subextensions of $N/K$ which are trivial in the Witt ring W$\left(K\right)$ of $K$. In particular, we prove that the torsion subgroup of $\mathcal{B}\left(\mathcal{G}\right)/\mathcal{T}\left(\mathcal{G}\right)$ coincides with the kernel of the total signature homomorphism.

We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14....