See the 1996 preprint by Vladimir Voevodsky and the wikipedia article.

- Fabien Morel,
*Voevodsky’s proof of Milnor’s conjecture*, Bull. Amer. Math. Soc. 35 (1998), 123-143 (web)

A self-contained proof of Milnor’s conjecture, and more generally the norm residue isomorphism theorem is given by

- Christian Haesemeyer? and Charles Weibel,
*The Norm Residue Theorem in Motivic Cohomology*(pdf)

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