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Unitary descent properties

Let k be a field of characteristic /=2 and let W(k) be the Witt ring of k and L a finite extension of k. If L/k is a Galois extension, then the image of rL/k is contained in W(L)Gal(L/k) where rL/k:W(k)→W(L) is the canonical ring homomorphism. Rosenberg and Ware (1970) proved that if L is a finite Galois extension of odd degree of k, then rL/k:W(k)→W(L)Gal(L/k) is an isomorphism. In this paper, the author generalizes the Rosenberg-Ware Theorem to the Witt group of division algebras with involution. Let A be a finite-dimensional k-algebra and σ:A→A a k-linear involution. Considering nondegenerate Hermitian forms over free (A,σ)-modules of finite rank, one obtains a Witt group W(A,σ). Let L be a field extension of k. In the main theorem (Theorem 1.2), the author proves that if L/k is a Galois extension of odd degree, then the canonical map rL/k:W(A,σ)→W(AL,σL)Gal(L/k) is an isomorphism, where AL=A⊗kL and σL is the extension of σ to AL. In Section 2, the author proves the main theorem, Theorem 1.2, by the method of Knebusch and Scharlau. She also extends a descent result of M. Rost [J. Ramanujan Math. Soc. 14 (1999), no. 1, 55--63; MR1700870 (2000f:11043)] concerning Witt groups in arbitrary odd degree extensions. Descent questions for Hermitian forms and their relations to isotropy properties are also discussed as well as descent in Galois cohomology. At the end of the paper, an application is made to bilinear forms invariant by finite groups.

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