A descent property for Pfister forms. (English) Zbl 1059.11033

Rost, Markus

J. Ramanujan Math. Soc. 14, No. 1, 55-63 (1999).

Summary: The Rosenberg-Ware theorem states that for a Galois extension \(K/F\) of odd degree the natural map of Witt rings of quadratic forms \[ W(F)\to W(K)^{\text{Gal}(K/F)} \] is an isomorphism. We extend this result to arbitrary field extensions \(K/F\) of odd degree. Basically we show that (Proposition 1) \[ 0\to W(F) @> r_{K/F} >> W(K) @> i_1-i_2 >> W(K\otimes K) \] is exact, where \(i_1,i_2\) are induced from the two natural maps \(K\to K\otimes K\). Further it is shown that an element of the graded Witt ring is equivalent to a Pfister form if this is true after an extension of odd degree (Proposition 2). We apply this to trace forms of exceptional Jordan algebras (Proposition 3). In the last section similar questions for symbols in Milnor’s \(K\)-theory and Galois cohomology are considered.

MSC:

11E81	Algebraic theory of quadratic forms; Witt groups and rings
19D45	Higher symbols, Milnor \(K\)-theory