Similarity of quadratic forms and related problems.

Abstract Let L / F be a field extension, φ , ψ even-dimensional quadratic forms over F. Assume that the forms φ L and ψ L are similar over L. In the first section we investigate whether it is possible to choose a similarity coefficient from F ⁎. The answer turns out to be positive if the degree of L / F is odd, and negative in general if L / F is a quadratic extension. In the second and third sections we apply the main theorem of the paper to the following problem: Let L (e) / L be a quadratic extension, e ∈ L ⁎ , Φ a form over L. Suppose that dim Φ is even and ind (Φ L (e)) = 1 2 ind (Φ). Does there exist a form Ψ over L such that Ψ L (e) is similar to Φ L (e) , disc (Ψ) = disc (Φ) , and ind (Ψ) = 1 2 ind (Φ) ? We show that the answer is negative in general almost for all conceivable pairs (dim Φ , ind (Φ)). On the other hand, the answer is positive if ind (Φ) = 2 , 4 ≤ dim Φ ≤ 8 , and if ind (Φ) = 4 , dim Φ = 6. In the fourth section we describe the set of similarity coefficients for pairs of 4-dimensional forms. Applying this description, we investigate the CV property with respect to quadratic extensions and pairs of forms (φ , ψ) , where φ and ψ are similar to 2-fold and 1-fold Pfister forms respectively. In the final section we establish general nontriviality of the quotient groups N F (e) / F F (e) ⁎ ∩ F ⁎ 2 Nrd (D F ⁎) F ⁎ 2 N F (e) / F (Nrd (D F (e) ⁎)) and N F (d) / F F (d) ⁎ ∩ N F (e) / F F (e) ⁎ ∩ D (π) F ⁎ 2 N F (d , e) / F (D (π F (d , e))) , where D ∈ Br 2 (F) , and π is a Pfister form over F. [ABSTRACT FROM AUTHOR]