Finding non-trivial elements in \(K_ 2({\mathcal O}_ F)\) for imaginary quadratic number fields \(F\)
Let \(F\) be a number field, \(S\) a finite set of primes containing the Archimedean ones, \({\mathcal O}_ F\) (resp. \({\mathcal O}_ S\)) the ring of integers (resp. \(S\)-integers) of \(F\). When \(F\) contains a primitive \(n\)th root of unity, there is a canonical map \(I: \mu_ n\otimes\text{Cl}({\mathcal O}_ S)\to K_ 2({\mathcal O}_ S)/n\) [\textit{F. Keune}, \(K\)-theory 2, 625-645 (1989; Zbl 0705.19007)]. If \(S\) contains the divisors of \(n\), \(I\) is injective and its image, roughly speaking, comes from the wild kernel (this could also be proved by means of Poitou- Tate's duality). If \(F\) does not contain enough roots of unity, then of course one can go up some suitable cyclotomic extension, and then go down by transfer. By using the map \(I\), the author proved in his thesis (Nijmegen, 1991) that if \(F\) contains ``many'' units, then \(K_ 2({\mathcal O}_ F)\) can be generated by Dennis-Stein symbols. In the opposite situation (e.g., when \({\mathcal O}^ x_ F\) is finite), it is even hard to point out non-trivial elements in \(K_ 2({\mathcal O}_ F)\). In this paper, the author gives a new construction of \(I\) which appeals to matrices and lends itself to calculation. He is then able to compute non-trivial elements in \(K_ 2({\mathcal O}_ F)\), \(F\) being an imaginary quadratic field.
scientific article; zbMATH DE number 166142
scientific article