Quadratic descent of involutions in degree 2 and 4

Dherte, Hélène

doi:10.1090/S0002-9939-1995-1243165-2

Quadratic descent of involutions in degree $2$ and $4$
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by Hélène Dherte

Proc. Amer. Math. Soc. 123 (1995), 1963-1969

DOI: https://doi.org/10.1090/S0002-9939-1995-1243165-2

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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 $A = A’ { \otimes _F}K$, 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.

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