mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' with coefficients in Z/2Z. It was proved by .

Statement

Let ''F'' be a field of characteristic different from 2. Then there is an isomorphism :

K_n^M(F)/2 \cong H_^n(F, \mathbb/2\mathbb)

for all ''n'' ≥ 0, where ''K^M'' denotes the

Milnor ring In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebr ...

About the proof

The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost,

Fabien Morel Fabien Morel (born 22 January 1965, in Reims) is a French algebraic geometer and key developer of A¹ homotopy theory with Vladimir Voevodsky. Among his accomplishments is the proof of the Friedlander conjecture, and the proof of the complex cas ...

, Eric Friedlander, and others, including the newly minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.

Generalizations

The analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky and Markus Rost yielded a complete proof of this conjecture in 2009; the result is now called the norm residue isomorphism theorem.

References

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Statement

About the proof

Generalizations

References

Further reading