Computation of \(K_ 2\mathbb{Z}[\sqrt {-6}]\)
Bass and Tate's method of computing \(K_ 2 O_ F\) consisting essentially in finding a finite set of primes \(S\) containing the set \(S_ \infty\) of archimedean primes, and such that: \[ K_ 2 O_ F= \text{Ker} (K_ 2^ S F\to \prod_{v\in S- S_ \infty} k(v)^*), \] where \(k(v)\) is the residue field and \(K_ 2^ S F\) is the subgroup of \(K_ 2 F\) generated by symbols of \(S\)-units. This method was carried out by Tate for all the Euclidean imaginary quadratic fields and for \(\mathbb{Q} (\sqrt{-15})\), using a series of technical lemmas. In the present paper, these lemmas are improved (also in a very technical way) and yield the triviality of \(K_ 2 O_ F\) for \(F= \mathbb{Q}( \sqrt{-6})\). The author also states that the same method allows him to prove that \(K_ 2 O_ F\) is trivial for \(F= \mathbb{Q}( \sqrt{-5})\) and \(\mathbb{Q}( \sqrt{-19})\) (Skalba's results) and \(K_ 2 O_ F\simeq \mathbb{Z}/ 2\mathbb{Z}\) for \(F= \mathbb{Q} (\sqrt{- 35})\) (to appear).
scientific article; zbMATH DE number 699138
scientific article