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Let F be a field of characteristic 2 and let K/F be a purely inseparable extension of exponent 1. We show that the extension is excellent for quadratic forms. Using the excellence, we recover and extend results by Aravire and Laghribi who computed generators for the kernel Wq(K/F) of the natural restriction map Wq(F) → Wq(K) between the Witt groups of quadratic forms of F and K, respectively, where K/F is a finite multiquadratic extension of separability degree at most 2. [ABSTRACT FROM AUTHOR]

We prove isomorphism criteria for Witt rings and reduced Witt rings of certain types of real fields. Refined criteria are obtained under the additional assumption that the field be SAP. This leads to a generalization of a result by Koprowski on Witt equivalence of function fields of transcendence degree 1 over a real closed field. Isomorphism criteria are also obtained for Witt rings of hermitian forms over a quadratic extension of a real base field and for Witt groups of hermitian forms over a quaternion algebra with a real field as center. All these criteria are expressed in terms of properties involving topological subspaces of the space of orderings of the base field. [ABSTRACT FROM AUTHOR]

Let F be a field and p a prime number. The p-symbol length of F, denoted by λp(F), is the least integer l such that every element of the group K2F/pK2F can be written as a sum of ≤ l symbols (with the convention that λp (F) = ∞ if no such integer exists). In this article, we obtain an upper bound for λp (F) in the case where the group F×/F×p is finite of order pm. This bound is λp(F) ≤ m/2, except for the case where p = 2 and F is real, when the bound is λp(F) ≤ m+1 / 2. We further give examples showing that these bounds are sharp. [ABSTRACT FROM AUTHOR]

Let WF denote the Witt ring of a field F of characteristic ≠2 and let I n F denote the n-th power of the ideal IF of even-dimensional forms in WF. The Arason-Pfister Hauptsatz states that if 0≠ϕ∈ I n F is anisotropic then dim ϕ≥ 2 n . Pfister also showed that if ϕ∈ I 3 F is anisotropic and dim ϕ>8 then dim ϕ≥12. We extend this result to I 4 F and show that if ϕ∈ I 4 F is anisotropic and dim ϕ>16 then dim ϕ≥24 and we provide some results on anisotropic 24-dimensional forms in I 4 F. [ABSTRACT FROM AUTHOR]