## $$K_ 3$$ of a field and the Bloch group.(English. Russian original)Zbl 0741.19005

Proc. Steklov Inst. Math. 183, 217-239 (1991); translation from Tr. Mat. Inst. Steklova 183, 180-199 (1990).
The author gives an exposition of his results, which were got mostly around 1982. The main problem is the computation of $$K_ 3(F)_{nd}=\text{Coker}(K^ M_ 3(F)\to K_ 3(F))$$, where $$F$$ is an infinite field, $$K^ M(F)$$ is the Milnor $$K$$-group and $$K(F)$$ is the Quillen $$K$$-group. The result is given in terms of the Bloch group $$B(F)$$ describing the non-trivial relations between tensor elements $$x\otimes(1- x)$$ in $$F^*\otimes F^*$$. It is proved that there is an exact sequence $0\to\widetilde{Tor}(F^*,F^*)\to K_ 3(F)_{nd}\to B(F)\to 0,$ where $$\widetilde{Tor}(F^*,F^*)$$ is the unique nontrivial extension of $$Tor(F^*,F^*)$$ by $$\mathbb{Z}/2$$.

### MSC:

 19D45 Higher symbols, Milnor $$K$$-theory 11R70 $$K$$-theory of global fields 19D55 $$K$$-theory and homology; cyclic homology and cohomology

### Keywords:

Milnor $$K$$-group; Quillen $$K$$-group; Bloch group