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This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays available.

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    $\begingroup$ Though I'm not at all a specialist in these matters, I wouldn't expect too much simplification: results of Garland and Tate have many connections to relatively deep theorems of number theory. Anyway, Garland's paper is A finiteness theorem for $K_2$ of a number field, Ann. of Math. (2) 94 (1971), 534–548. The Bourbaki Seminar report by Bass is freely available: numdam.org/numdam-bin/fitem?id=SB_1970-1971__13__233_0 $\endgroup$
    – Jim Humphreys
    Commented Oct 25, 2014 at 19:16

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For a calculation of $K^M_2(K)$ for $K = \mathbf{Q}$, see Lemmermeyer, Reciprocity Laws, p. 66, Theorem 2.30. (Note that the direct product should be a direct sum.)

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