Formes Quadratiques de Dimension 6

The length and other invariants of a real field

“…16 We start with constructing a real field k with elements a, b ∈ k 2 such that the quadratic form 1, 1, 1, a, −b, ab over k is anisotropic. To this end we take a, c ∈ N such that a ≡ 1 mod 8, c ≡ 7 mod 8, and a is not a square number (e.g., a = 17 and c = 7).…”

Bilinear forms of dimension less than or equal to 5 and function fields of quadrics in characteristic 2

“…The isotropy problem was considered first in the case where F is of characteristic not 2 and φ is of dimension ≤8. Many persons contributed in this case: Shapiro 31, Leep 28, Merkurjev 26, Hoffmann 3, 4, 6, the first author 15–17, Karpenko and Izhboldin 10–13 (probably our list is not exhaustive neither for the results proved about this problem in characteristic not 2, nor for the authors contributing to this problem). It is the work of Merkurjev in the nineties on the u ‐invariant 26, 29 which gave a revival to this problem.…”

Some new examples in the theory of quadratic forms

“…In the case dim φ ≤ 5, a complete classification is known (see [11]). The case dim φ = 6 was studied in [12], [31], [32], [34], and [37]. In the case where dim φ = 6 and dim ψ = 4, a complete classification was obtained.…”