Generalization of Thue's theorem and computation of the group \(K_ 2 O_ F\) (Q1323875)

The aim of this paper is to present a certain adaptation of Bass and Tate's method of computing \(K_ 2\) of a number field \(F\). It is based on an extension to the ring of integers \(O_ F\) of a theorem of Thue concerning linear congruences in \(\mathbb{Z}\). This improves the ``geometry of numbers'' part in Bass and Tate's method and can be applied to give a new proof of the finite generation of \(K_ 2 O_ F\) (identified with the tame kernel). With suitable minor modifications, this also yields the triviality of \(K_ 2 O_ F\) for certain imaginary quadratic fields, such as \(\mathbb{Q} (\sqrt{-19})\) (detailed proof) or \(\mathbb{Q} (\sqrt{-5})\) (sketched proof).