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| In [[mathematics]], [[Helmut Hasse]]'s '''local-global principle''', also known as the '''Hasse principle''', is the idea that one can find an [[diophantine equation|integer solution to an equation]] by using the [[Chinese remainder theorem]] to piece together solutions [[modular arithmetic|modulo]] powers of each different [[prime number]]. This is handled by examining the equation in the [[Completion_(ring_theory)|completions]] of the [[rational number]]s: the [[real number]]s and the [[p-adic number|''p''-adic numbers]]. A more formal version of the Hasse principle states that certain types of equations have a rational solution [[if and only if]] they have a solution in the [[real number]]s ''and'' in the ''p''-adic numbers for each prime ''p''.
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| | If you've got to reload their arms when playing conflict of clans hack and that's shooting entailed, always get cover first. It truly is common for suppliers to be [http://mondediplo.com/spip.php?page=recherche&recherche=gunned+low gunned low] while a reload is happening, and you watches helplessly. Do And never let it happen you r! Find somewhere to conceal before you start up to reload.<br><br>Stay aware of how multi-ply player works. In the instance you're investing in the best game exclusively for its multiplayer, be sure the person have everything required needed for this. If you really are planning on playing a person in your prized household, you may ascertain that you will have two copies of the clash of clans cheats to game against one another.<br><br>Clash of Clans is obviously a popular sport designed to end up being played on multiple systems, the car . iOS and also android. The overall game is terribly intriguing but presently presently comes a spot each morning legend, where the individual gets trapped because involved with not enough gems. However, this problem is becoming able to easily be resolved.<br><br>Provde the in-online game songs possibility. If, nonetheless, you might nevertheless be annoyed by using this tool soon after one time approximately, don't be scared to mute the television set or personal computer and as a consequence play some audio of your very own. You'll find a far more satisfying game playing experience performing this and therefore are a good deal more unlikely to get a huge frustration from actively collaborating in.<br><br>Make sure that the customer build and buy new laboratory so you're able to research improved barbarians. If you beloved this informative article as well as you want to receive more details relating to [http://circuspartypanama.com clash of clans hack android] kindly visit the web page. Eventually, in casing you take part involving game for most months, you might finally manage to get thier the nirvana of five-star barbarians.<br><br>A leap into the pre-owned or operated xbox gaming marketplace. Several fans will get a Clash of Clans Hack and complete this game really conveniently. Several shops let these gaming titles being dealt in and then also promote them at your lessened cost. You might be by far probably the most cost-effective technique to get newer video games the particular higher cost. |
| Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and ''p''-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution?
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| One can ask this for other rings or fields: integers, for instance, or [[number field]]s. For number fields, rather than reals and ''p''-adics, one uses complex embeddings and <math>\mathfrak p</math>-adics, for [[prime ideal]]s <math>\mathfrak p</math>.
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| ==Forms representing 0==
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| ===Quadratic forms===
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| The [[Hasse–Minkowski theorem]] states that the local-global principle holds for the problem of representing 0 by [[quadratic form]]s over the [[rational number]]s (which is [[Hermann Minkowski|Minkowski]]'s result); and more generally over any [[number field]] (as proved by Hasse), when one uses all the appropriate [[local field]] necessary conditions. [[Hasse's theorem on cyclic extensions]] states that the local-global principle applies to the condition of being a relative norm for a cyclic extension of number fields.
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| ===Cubic forms===
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| A counterexample by [[Ernst S. Selmer]] shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3''x''<sup>3</sup> + 4''y''<sup>3</sup> + 5''z''<sup>3</sup> = 0 has a solution in real numbers, and in all p-adic fields, but it has no solution in which ''x'', ''y'', and ''z'' are all rational numbers.<ref>{{cite journal | author=Ernst S. Selmer | title=The Diophantine equation ''ax''<sup>3</sup> + ''by''<sup>3</sup> + ''cz''<sup>3</sup> = 0 | journal=Acta Mathematica | volume=85 | pages=203–362 | year=1957 | doi=10.1007/BF02395746 }}</ref>
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| [[Roger Heath-Brown]] showed<ref>{{cite journal | author=D.R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in 14 variables | journal=Invent. Math. | volume=170 | pages=199–230 | year=2007 | doi=10.1007/s00222-007-0062-1}}</ref> that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of [[Harold Davenport|Davenport]].<ref>{{cite journal | author=H. Davenport | title=Cubic forms in sixteen variables | journal=[[Proceedings of the Royal Society A]] | volume=272 | pages=285–303 | year=1963 | doi=10.1098/rspa.1963.0054 | issue=1350 }}</ref> Hence the local-global principle holds trivially for cubic forms over the rationals in at least 14 variables.
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| If we confine ourselves to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0,<ref>{{cite journal | author=D. R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in ten variables | journal=Proceedings of the London Mathematical Society| volume=47 | pages=225–257 | year=1983 | doi=10.1112/plms/s3-47.2.225 | issue=2 }}</ref> thus trivially establishing the Hasse principle for this class of forms. It is known that Heath-Brown's result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that don't represent zero.<ref>{{cite journal | author=L. J. Mordell | authorlink=Louis Mordell | title=A remark on indeterminate equations in several variables | journal=Journal of the London Mathematical Society | volume=12 |pages=127–129 | year=1937 | doi=10.1112/jlms/s1-12.1.127 | issue=2 }}</ref> However, [[Christopher Hooley|Hooley]] showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables.<ref>{{cite journal | author=C. Hooley | authorlink=Christopher Hooley | title=On nonary cubic forms | journal=J. Für die reine und angewandte Mathematik | volume=386 |pages=32–98 | year=1988 }}</ref> Davenport, Heath-Brown and Hooley all used the [[Hardy–Littlewood circle method]] in their proofs. According to an idea of [[Yuri Ivanovitch Manin|Manin]], the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the [[Brauer group]]; this is the [[Brauer–Manin obstruction]], which accounts completely for the failure of the Hasse principle for some classes of variety. However, [[Alexei Skorobogatov|Skorobogatov]] has shown that this is not the complete story.<ref>{{cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=Invent. Math. | volume=135 | issue=2 | pages=399–424 | year=1999 | doi=10.1007/s002220050291 }}</ref> | |
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| ===Forms of higher degree===
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| Counterexamples by [[Masahiko Fujiwara|Fujiwara]] and [[Masaki Sudo|Sudo]] show that the Hasse–Minkowski theorem is not extensible to forms of degree 10''n'' + 5, where ''n'' is a non-negative integer.<ref>{{cite journal | author=M. Fujiwara | authorlink=Masahiko Fujiwara | coauthors=[[Masaki Sudo|M. Sudo]] | title=Some forms of odd degree for which the Hasse principle fails | journal=Pacific Journal of Mathematics | volume=67 | year=1976 | issue=1 | pages=161–169}}</ref>
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| On the other hand, [[Birch's theorem]] shows that if ''d'' is any odd natural number, then there is a number ''N''(''d'') such that any form of degree ''d'' in more than ''N''(''d'') variables represents 0: the Hasse principle holds trivially.
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| ==Albert–Brauer–Hasse–Noether theorem==
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| The [[Albert–Brauer–Hasse–Noether theorem]] establishes a local-global principle for the splitting of a [[central simple algebra]] ''A'' over an algebraic number field ''K''. It states that if ''A'' splits over every [[local field|completion]] ''K''<sub>''v''</sub> then it is isomorphic to a [[matrix ring|matrix algebra]] over ''K''.
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| ==Hasse principle for algebraic groups==
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| The Hasse principle for algebraic groups states that if ''G'' is a simply-connected algebraic group defined over the global field ''k'' then the map from
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| :<math> H^1(k,G)\rightarrow\prod_s H^1(k_s,G)</math> | |
| is injective, where the product is over all places ''s'' of ''k''.
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| The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
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| {{harvtxt|Kneser|1966}} and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group ''E''<sub>8</sub> which was only completed by {{harvtxt|Chernousov|1989}} many years after the other cases.
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| The Hasse principle for algebraic groups was used in the proofs of the [[Weil conjecture for Tamagawa numbers]] and the [[strong approximation theorem]].
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| ==See also==
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| * [[Local analysis]]
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| * [[Grunwald–Wang theorem]]
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| * [[Grothendieck–Katz p-curvature conjecture]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{citation|last= Chernousov|first= V. I. |title=The Hasse principle for groups of type E8 |journal= Soviet Math. Dokl. |volume= 39 |year=1989|pages= 592–596|mr= 1014762}}
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| *{{Citation | last1=Kneser | first1=Martin | title=Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | id={{MR|0220736}} | year=1966 | chapter=Hasse principle for H¹ of simply connected groups | pages=159–163}}
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| * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=250–258 }}
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| * {{cite book | title=Torsors and rational points | author=Alexei Skorobogatov | series=Cambridge Tracts in Mathematics | volume=144 | year=2001 | isbn=0-521-80237-7 | pages=1–7,112 | publisher=Cambridge Univ. Press | location=Cambridge }}
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| == External links ==
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| * {{springer|title=Hasse principle|id=p/h046670}}
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| * [http://planetmath.org/encyclopedia/HassePrinciple.html PlanetMath article]
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| * Swinnerton-Dyer, ''Diophantine Equations: Progress and Problems'', [http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf online notes]
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| [[Category:Algebraic number theory]]
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| [[Category:Diophantine equations]]
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| [[Category:Localization (mathematics)]]
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| [[Category:Mathematical principles]]
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