In mathematics, the Milnor conjecture was a proposal by of a description of the

Milnor K-theory 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 ...

(mod 2) of a general

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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 Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic var ...

uses several ideas developed by Voevodsky,

Alexander Merkurjev Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev ...

Andrei Suslin Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Truste ...

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. He is a professor at the University of Bielefeld. ...

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 ca ...

Eric Friedlander Eric Mark Friedlander (born January 7, 1944 in Santurce, Puerto Rico) is an American mathematician who is working in algebraic topology, algebraic geometry, algebraic K-theory and representation theory. Friedlander graduated from Swarthmore ...

, and others, including the newly minted theory of

motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geome ...

(a kind of substitute for

singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

for

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...

) and the

motivic Steenrod algebra In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The motiv ...

Generalizations

The analogue of this result for

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky and

yielded a complete proof of this conjecture in 2009; the result is now called the

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 o ...

References

* * * * *

Statement

About the proof

Generalizations

References

Further reading