On Hilbert Satz 90 for K₃ for degree-two extensions

by Markus Rost (Preprint, May 86, 14 pages)

The text describes a proof of Hilbert's Satz 90 for K₃ for quadratic extensions.

The results had been obtained independently by A. Merkurjev and A. Suslin, see

• Merkurjev, A. S., Suslin, A. A, Norm residue homomorphism of degree three. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 339-356; translation in Math. USSR Izv. 36 (1991), no. 2, 349-367. MR 1062517.
• Merkurjev, A. S., Suslin, A. A, The group K₃ for a field. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 522-545; translation in Math. USSR Izv. 36 (1991), no. 3, 541-565. MR 1072694.

The two proofs differ significantly concerning the investigation of the residue maps on K_3 of the function field of a conic. While Merkurjev and Suslin use here their general results about indecomposable K_3, our method consists in specific considerations of conics. We use Rehmann's description of K₂ of skew fields and the Reidemeister-Schreier method to obtain a rational parametrization of generators and relations for K₂ of a quaternion algebra.

For the final conclusion one uses also the preprint Injectivity of K₂(D) -> K₂(F) for quaternion algebras. An "elementary" proof of the result can be found in the preprint On Hilbert Satz 90 for K₃ for quadratic extensions.

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