The Structure of Certain K2OF

Abstract:

      In this paper, we investigate the structure of K₂O_F for F = Q(d^1/2), d ≡ -3 mod 9 and d ≠ -3. We find the element of order 3 of K₂O_F for F = Q((-21)^1/2) and generated elements of K₂O_F≌z/(2)⊕z/(8)⊕z/(3) for F = Q((15)^1/2). We get the property of R₂F, which develops a Tate and Bass's theorem, and give the structure of K₂O_F for F = Q((29)^1/2) and the presentation relations of SL_n(O_F)(n≥ 3).