In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Milnor conjecture was a proposal by of a description of the Milnor K-theory

Milnor K-theory

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by .The calculation of K2 of a field k led Milnor to the following ad hoc definition of "higher" K-groups by...

 (mod 2) of a general field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 F with characteristic

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

 different from 2, by means of the Galois

Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...

 (or equivalently étale

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

) cohomology of F with coefficients in Z/2Z. It was proved by .

Statement of the theorem

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


for all n ≥ 0.

About the proof

The proof of this theorem by Vladimir Voevodsky

Vladimir Voevodsky

Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...

 uses several ideas developed by Voevodsky, Andrei Suslin

Andrei Suslin

Andrei Suslin is a Russian mathematician who has made major contributions to the field of algebra, especially algebraic K-theory and its connections with algebraic geometry. He is currently a Trustee Chair and Professor of mathematics at Northwestern University.He was born on December 27, 1950,...

, Fabien Morel, Eric Friedlander, and others, including the newly-minted theory of motivic cohomology

Motivic cohomology

Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...

 (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.

Generalizations

The analogue of this result for prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky, Markus Rost

Markus Rost

Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China...

, and Charles Weibel yielded a complete proof of this conjecture in 2009; the result is now called the norm residue isomorphism theorem

Norm residue isomorphism theorem

In the mathematical field of algebraic K-theory, the norm residue isomorphism theorem is a long-sought result whose complete proof was announced in 2009...

.