Norm residue isomorphism theorem

In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime ${\displaystyle \ell }$ and any natural number ${\displaystyle n}$. John Milnor^[1] speculated that this theorem might be true for ${\displaystyle \ell =2}$ and all ${\displaystyle n}$, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato^[2] and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions.^[3] The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.