Quadratic descent of involutions in degree 2 and 4

Abstract

Abstract.If K/F is a quadratic extension, we give necessary and sufficient conditions in terms of the discriminant (resp. the Clifford algebra) for a quadratic form of dimension 2 (resp. 4) over K to be similar to a form over F . We give similar criteria for an orthogonal involution over a central simple algebra A of degree 2 (resp. 4) over K to be such that A = A' ®F K , where A' is invariant under the involution. This leads us to an example of a quadratic form over K which is not similar to a form over F but such that the corresponding involution comes from an involution defined over F .

Quadratic descent of totally decomposable orthogonal involutions in characteristic two

“…Write L = F [v 1 , · · · , v r ] for some v 1 , · · · , v r ∈ L. Then E ≃ K[v 1 , · · · , v r ]. Since Φ(A, σ) is generated, as a K-algebra, by n elements, by [7, (3.9)] there exist v r+1 , · · · , v n ∈ Φ(A, σ) such that Φ(A, σ) = K[v 1 , · · · , v n ] and v 2 i = 0, i = r + 1, · · · , n. This proves parts (1) and (2). The third part follows by replacing v i with v i +1 for i = r+1, · · · , n.…”

“…i.e., u i w i = w i u i . Since g(v i ) 2 ∈ K, (2) and (3) imply that u 2 i ∈ F for every i. We now show that w i w j = w j w i for i, j = 1, · · · , n. As (6.6) implies that (w i + w j ) 2 = α i + α j , i.e., w i w j = w j w i .…”

Quadratic descent of hermitian forms

“…It follows that has a descent to F if and only if splits. (ii) If , the claim follows from [, (2.4)] and its proof. Otherwise, it can be found in [, (4.7)].…”

Descent properties of biquaternion algebras with involution

“…If in addition , we say that has a quadratic descent to K . In characteristic different from two, the quadratic descent of was studied in for the case where σ is orthogonal and the degree of A is at most 4. Also, the quadratic descent of totally decomposable algebras with orthogonal involution in characteristic two was recently studied in ; see also [, (7.6)].…”

“…For example, some descent properties of a quadratic form over the function field of a K ‐quadric were studied in and . Also, the quadratic descent of quadratic forms of dimension 2 and 4 in characteristic different from 2 was investigated in .…”