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, Volume 24, Issue 2, pp 173–177 | Cite as

The elements of K2(ℤs)

  • Jürgen Hurrelbrink


Let S={p1,...,ps} be a set of rational primes,\(\mathbb{Z}_S : = \mathbb{Z}[\frac{1}{{P_1 }},...,\frac{1}{{P_5 }}]\). One has K2(ℤs)≅K2(ℤsu{2} and we want to assume 2 ∈ S. It is snown that every element of K2(ℤS) is a Dehnis-Stein-symbol <a,b>, 1+ab being a unit of ℤS.

Here b ∈ ℤ can be determined concretely, depending only on S, and we obtain a normal form of the elements of K2(Q) as Steinberg-symbols, which is unique in some way and expresses the quadratic reciprocity law.


Normal Form Number Theory Algebraic Geometry Topological Group Quadratic Reciprocity 
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  1. [1]
    DENNIS, R.K., STEIN, M.R.: The functor K2: a survey of computations and problems. Algebraic K-Theory II. Springer Lect. Notes in Math.342, 243–280 (1973)Google Scholar
  2. [2]
    LENSTRA, H. W. Jr.: K2 of a global field consists of symbols. Algebraic K-Theory. Springer Lect. Notes in Math.551, 69–73, (1976)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Jürgen Hurrelbrink
    • 1
  1. 1.Fakultät für MathematikBielefeld 1

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