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| {{for|Milnor's conjecture about the slice genus of torus knots|Milnor conjecture (topology)}}
| | Nice to meet you, I am Marvella Shryock. I used to be unemployed but now I am a librarian and the salary has been truly satisfying. One of the issues she loves most is to do aerobics and now she is trying to make cash with it. California is our birth place.<br><br>Feel free to visit my web-site ... std testing at home; [http://premium.asslikethat.com/user/CSalerno the full report], |
| In [[mathematics]], the '''Milnor conjecture''' was a proposal by {{harvs|txt|first=John|last= Milnor|year=1970|authorlink=John Milnor}} of a description of the [[Milnor K-theory]] (mod 2) of a general [[field (mathematics)|field]] ''F'' with [[characteristic (algebra)|characteristic]] different from 2, by means of the [[Galois cohomology|Galois]] (or equivalently [[étale cohomology|étale]]) cohomology of ''F'' with coefficients in '''Z'''/2'''Z'''. It was proved by {{harvs|txt|authorlink=Vladimir Voevodsky|first=Vladimir|last= Voevodsky|year1=1996|year2=2003a|year3=2003b}}.
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| ==Statement of the theorem==
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| Let ''F'' be a field of characteristic different from 2. Then there is an isomorphism
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| :<math>K_n^M(F)/2 \cong H_{\acute{e}t}^n(F, \mathbb{Z}/2\mathbb{Z})</math>
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| for all ''n'' ≥ 0, where ''K'' denotes the [[Milnor ring]].
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| ==About the proof==
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| The proof of this theorem by [[Vladimir Voevodsky]] uses several ideas developed by Voevodsky, [[Alexander Merkurjev]], [[Andrei Suslin]], [[Markus Rost]], [[Fabien Morel]], [[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]].
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| ==Generalizations==
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| The analogue of this result for [[prime number|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]].
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| ==References==
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| * {{Citation | last1=Mazza | first1=Carlo | last2=Voevodsky | first2=Vladimir | author2-link=Vladimir Voevodsky | last3=Weibel | first3=Charles | title=Lecture notes on motivic cohomology | url=http://math.rutgers.edu/~weibel/motiviclectures.html | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Clay Mathematics Monographs | isbn=978-0-8218-3847-1 | mr=2242284 | year=2006 | volume=2 | author-link=Charles Weibel}}
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| * {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Algebraic K-theory and quadratic forms | doi=10.1007/BF01425486 | mr=0260844 | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=9 | pages=318–344 | issue=4}}
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| * {{citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | url=http://www.math.uiuc.edu/K-theory/0170 | title=The Milnor Conjecture | year=1996 | series=Preprint}}
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| * {{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Reduced power operations in motivic cohomology | url=http://www.numdam.org/item?id=PMIHES_2003__98__1_0 | doi=10.1007/s10240-003-0009-z | mr=2031198 | year=2003a| journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | issue=98 | pages=1–57 | volume=98}}
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| * {{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Motivic cohomology with Z/2-coefficients | url=http://www.numdam.org/item?id=PMIHES_2003__98__59_0 | doi=10.1007/s10240-003-0010-6 | mr=2031199 | year=2003b | journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | issue=98 | pages=59–104 | volume=98}}
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| ==Further reading==
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| * {{citation | last=Kahn | first=Bruno | chapter=La conjecture de Milnor (d'après V. Voevodsky) | language=French | editor1-last=Friedlander | editor1-first=Eric M. | editor2-last=Grayson | editor2-first=D.R. | title=Handbook of ''K''-theory | volume=2 | pages=1105–1149 | publisher=[[Springer-Verlag]] | year=2005 | isbn=3-540-23019-X | zbl=1101.19001 }}
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| [[Category:K-theory]]
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| [[Category:Conjectures]]
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| [[Category:Theorems in abstract algebra]]
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Nice to meet you, I am Marvella Shryock. I used to be unemployed but now I am a librarian and the salary has been truly satisfying. One of the issues she loves most is to do aerobics and now she is trying to make cash with it. California is our birth place.
Feel free to visit my web-site ... std testing at home; the full report,