Applications of Conics to Quadratic forms Over the Rational Function Fields

We show that for any forms ϕ₁ and ϕ₂ over a field k of characteristic different from 2 and a ∈ k∗, the anisotropic part of the form ϕ₁ ⊥ (t² − a)ϕ₁ over the rational function field k(t) is of the same type, i.e. there exist forms τ₁ and τ₂ over k such that (ϕ₁ ⊥ (t¹ − a)ϕ₂)an τ₁ ⊥ (t² − a)τ₂. Also we determine the structure of certain Pfister forms over k(t), and describe the behavior of quadratic forms under biquadratic extensions of k in terms of some related forms over the function field of the product of two conics over k(x), or k(x, y). The excellence property of the function field of a conic plays the central role throughout the paper.