24 - SK₁(D) ≠ 1 for suitable D

Summary

Until 1975 no example for SK₁(D) ≠ {1} for a suitable skew field D was known; the problem of finding one (or alternatively proving that there is none) often was referred to as the Tannaka–Artin Problem (particularly in the Russian literature).

Looking back all this seems strange since we shall introduce later in this paragraph such an example which – in order to be understood – requires hardly more knowledge on skew fields than displayed already in §1.!

The first example for SK₁(D) ≠ {1}, however, was somewhat more complicated; it was given by В.П. Платонов [1975]. Further information on that V.P. Platonov developed a whole theory of examples and published it in many (mostly short) papers) and related points of view (partially due to the author) may be obtained from the report P. Draxl & M. Kneser [1980] and the literature list therein (cf. also the remarks at the end of this paragraph).

Now for the just mentioned example: we start with the obvious

Lemma 1.If in Definition 1 in §1. the field L is assumed to be only a skew field (and not a commutative field), then Definition 1 in §1. still makes sense and Lemma 3 in §1. remains correct.