the next major development in the subject came with the definition of "k". steinberg studied the universal central extensions of a chevalley group over a field and gave an explicit presentation of this group in terms of generators and relations. in the case of the group e("k") of elementary matrices, the universal central extension is now written st("k") and called the "steinberg group". in the spring of 1967, john milnor defined "k"("r") to be the kernel of the homomorphism . the group "k" further extended some of the exact sequences known for "k" and "k", and it had striking applications to number theory. hideya matsumoto's 1968 thesis showed that for a field "f", "k"("f") was isomorphic to: