Determination of _{}(≥3) for global fields

A short proof is obtained for Tate's theorem about kn of global fields. For a field F, the groups KnF and knF=KnF/2KnF are first defined by Milnor [1]. Following the notations there, we shall write K1F= {l(a): aO} where l(ab)=l(a)+l(b). In this note, we present a simple proof of the following theorem of Tate (see Appendix of [1]): THEOREM. If F is a globalfield, and {F,} is thefamily of real completions of F, then for n >3, knF--?, knF, is an isomorphism. In particular, the dimension of kJF as a Z2-vector space is equal to the number of real completions of F.2 We shall need two lemmas. LEMMA 1. If x,y andz=x+y are nonzero in F, then l(x)l(y)=I(z)l(-xy) in k2F. PROOF. Since xz-'+yz'-=1, we have l(xz-')l(yz-')=O by definition of K2(F). Therefore in k2F, (I(x) + l(z))(l(y) + I(Z)) = 0. This implies l(x)l(y) = l(z)l(z) + I(z)l(xy) = l(z)l(1) + I(z)l(xy) = l(z)l(-xy) in k2F. REMARK. The same proof shows that l(x)l(y)=l(z)l(-yx-1) in K3F. LEMMA 2. For nonzero a, b, c in F, l(a)l(b)l(c)=O in k3F unless there exists a real completion of F at which a, b, c are all negative. Received by the editors February 16, 1971. AMS 1970 subject classifications. Primary 18F25, Secondary 13D15, 12-00.