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Submission: 2013, Oct 5
Let q be a quadratic form over a field k and let L be a field extension of k of odd degree. It is a classical result that if q is isotropic (resp. hyperbolic) over L then q is isotropic (resp. hyperbolic) over k. In turn, given two quadratic forms q and q' over k, if q and q' are isomorphic over L then they are isomorphic over k. It is natural to ask whether similar results hold for algebras with involution. We give a survey of the progress on these three questions with particular attention to the relevance of hyperbolicity, isotropy and isomorphism over an appropriately defined function field. Incidentally, we prove the anisotropy property in some new low degree cases.
2010 Mathematics Subject Classification: 11E72
Keywords and Phrases: Springer's theorem, generic splitting, odd-degree extensions, involutions, isotropy, hyperbolicity, isomorphism
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