Algebraic $K$-theory and the norm residue homomorphism

Recent results on the structure of the group $K_2$ of a field and its connections with the Brauer group are presented. The $K$-groups of Severi–Brauer varieties and simple algebras are computed. A proof is given of Milnor's conjecture that for any field $F$ and natural number $n>1$ there is the isomorphism $R_{n,F}\colon K_2(F)/nK_2(F)\overset\sim\to_n\mathrm{Br}(F)$. Algebrogeometric applications of the main results are presented.