Computation of \(K_ 2\) for the ring of integers of quadratic imaginary fields.
For a set of finite places, \(S\), of a number field \(F\) let \(K_2^S(F)\) be the subgroup of \(K_2(F)\) generated by symbols of the form \(\{a,b\}\) with \(a,b\in U_S= \{x\in F^*\mid v(x)\geq 0\) \(\forall v\not\in S\}\). For imaginary quadratic fields \textit{J. Tate}, in the Appendix to his paper with \textit{H. Bass} on `The Milnor ring of a global field' [Lect. Notes Math. 342, 429--446 (1973; Zbl 0299.12013)] showed how to determine a bound \(M= M(F)\in \mathbb{N}\), such that \(K_2(O_F) \subset K_2^S(F)\) whenever \(S\) contains all finite places of norm at most \(M\). The authors improve Tate's bound (by refining Tate's methods) for certain quadratic imaginary fields and use the improved bounds to show that the group \(K_2(O_F)\) is trivial when \(F= \mathbb{Q} (\sqrt {-43})\).
scientific article; zbMATH DE number 1782130