The norm theorem for semisingular quadratic forms

The excellence of function fields of conics

“…This theorem played a crucial role for the descent of bilinear forms recently studied by the two authors [11]. Our proof of this theorem is inspired from Rost's proof in characteristic not 2, but we will adapt many arguments to our situation.…”

“…Recently the authors studied the excellence of quartic extensions. Among others, they proved that a mixed biquadratic extension is excellent for quadratic forms [12]. It is not difficult to see that this extension is also excellent for bilinear forms.…”

The behavior of singular quadratic forms under purely inseparable extensions

“…We divide this proof into two major steps: We first study the case where the semisingular form φ is of the type (1, s) and becomes quasi-hyperbolic over F ( 2 n √ d) (Proposition 5.2). In this first step, we use an induction on n, and we also take help of a recent result of ours [15,Theorem 1.1], which gives us that φ represents the polynomial x 2 n + d up to a scalar represented by φ. Further, we use the Cassel-Pfister theorem (Proposition 4.2).…”

“…2 + b)φ over F (y)[15, Proposition 5.3]. Moreover, the polynomialay 2 + b is a norm of ⟨1, a⟩ b ⊗ [1, bα −2 ]as it is represented by this form.…”