×

\(K_ 2\)-analogs of Hasse’s norm theorems. (English) Zbl 0539.12007

The classical norm theorems of Hasse for fields and of Hasse-Schilling for simple algebras can be stated in the framework of algebraic K-theory in terms of the functor \(K_ 1\). The authors prove analogs of these results for the functor \(K_ 2\). These analogs are deduced from the following theorem: Let K be a global field and L a finite extension of K. An element of \(K_ 2(K)\) lies in the image of the transfer homomorphism from \(K_ 2(L)\) to \(K_ 2(K)\) if and only if the norm residue symbols vanish at those real primes of K which have only complex extensions to L. In particular it turns out that the \(K_ 2\)-analog of Hasse’s norm theorem is valid for all finite extensions of K.
The proofs use deep number theoretic properties of the functor \(K_ 2\) and the existence of a reduced norm homomorphism for \(K_ 2\), established by A. S. Merkur’ev and A. A. Suslin [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)].
Reviewer: M.Kolster

MSC:

11R70 \(K\)-theory of global fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11R37 Class field theory
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11S45 Algebras and orders, and their zeta functions
11S31 Class field theory; \(p\)-adic formal groups
11R34 Galois cohomology

Citations:

Zbl 0525.18008
PDFBibTeX XMLCite
Full Text: DOI EuDML