en>LilHelpa: Typo fixing and general fixes using AWB

2014-11-02T18:12:30Z

Typo fixing and general fixes using AWB

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	~~:''There also~~ is ~~[[Brauer~~'s ~~theorem on induced characters]]~~.''		Oscar is how he's called and he completely enjoys this name. One of the extremely very best issues in the globe for him is to collect badges but he is struggling to find time for it. Managing individuals has been his day job for a while. Her husband and her reside in Puerto Rico but she will have to move 1 working day or another.<br><br>Look into my page ... [http://bikedance.com/blogs/post/15246 std testing at home]

	~~In [[mathematics]], '''Brauer's theorem''', named for [[Richard Brauer]], is a result on~~ the ~~representability of 0 by forms over certain [[field (mathematics)\|fields]]~~ in ~~sufficiently many variables.<ref>R. Brauer, ''A note on systems of homogeneous algebraic equations'', Bulletin of~~ the ~~American Mathematical Society, '''51''', pages 749-755 (1945)</ref>~~

	~~==Statement of Brauer's theorem==~~
	~~Let ''K'' be a field such that~~ for ~~every integer ''r'' > 0 there exists an integer ψ(''r'') such that~~ for ~~''n'' ≥ ψ(r) every equation~~

	~~:<math>(*)\qquad a_1x_1^r+\cdots+a_nx_n^r=0,\quad a_i\in K,\quad i=1,\ldots,n</math>~~

	has a ~~non-trivial (i.e. not all ''x''<sub>''i''</sub> are equal to 0) solution in ''K''.~~
	Then, given homogeneous polynomials ''f''<sub>1</sub>,...,''f''<sub>''k''</sub> of degrees ''r''<sub>1</sub>,...,''r''<sub>''k''</sub> respectively with coefficients in ''K'', for every set of positive integers ''r''<sub>1</sub>,.~~..,''r''<sub>''k''</sub>~~ and every non-negative integer ''l'', there exists a number ω(''r''<sub>1</sub>,...,''r''<sub>''k''</sub>,''l'') such that for ''n'' ≥ ω(''r''<sub>1</sub>,...,''r''<sub>''k''</sub>,''l'') there exists an ''l''-dimensional [[affine subspace]] ''M'' of ''K<sup>n</sup>'' (regarded as a vector space over ''K'') satisfying

	~~:<math>f_1(x_1,\ldots,x_n)=\cdots=f_k(x_1,\ldots,x_n)=0,\quad\forall(x_1,\ldots,x_n)\~~in ~~M.</math>~~

	~~==An application~~ to ~~the field of p-adic numbers==~~
	Letting ''K'' be the field of [[p-adic number]]s in the theorem, the equation () is satisfied, since <math>\mathbb{Q}_p^/\left(\mathbb{Q}_p^*\right)^b</math>, ''b'' a natural number, is finite. Choosing ''k'' = 1, one obtains the following corollary:

	~~:A homogeneous equation ''f''(''x''<sub>~~1~~</sub>,..~~.~~,''x''~~<~~sub~~>~~''n''~~<~~/sub~~>~~) = 0 of degree ''r'' in the field of p-adic numbers has a non-trivial solution if ''n'' is sufficiently large.~~

	One can show that if ''n'' is sufficiently large according to the above corollary, then ''n'' is greater than ''r''<sup>2</sup>. Indeed, [[Emil Artin]] conjectured<ref>''Collected papers of Emil Artin'', page~~ x, Addison–Wesley, Reading, Mass~~.~~, 1965</ref> that every homogeneous polynomial of degree ''r'' over '''Q'''<sub>''p''</sub> in more than ''r''<sup>2</sup> variables represents 0~~. ~~This is obviously true for ''r'' = 1, and it is well known that the conjecture is true for ''r'' = 2 (see, for example, J~~.~~-P. Serre, ''A Course in Arithmetic'', Chapter IV, Theorem 6). See~~ [~~[quasi-algebraic closure]] for further context.~~

	In 1950 Demyanov<ref>{{cite journal\| last=Demyanov \| first=V. B. \| year=1950 \|title=На кубических форм дискретных линейных нормированных полей \|trans_title=On cubic forms over discrete normed fields \| journal=[[Doklady Akademii Nauk SSSR]] \| volume=74 \| pages=889–891}}</ref> verified the conjecture for ''r'' = 3 and ''p'' ≠ 3, and in 1952 [[D. J. Lewis]]<ref>D. J. Lewis, ''Cubic homogeneous polynomials over p-adic number fields'', Annals of Mathematics, '''56''', pages 473–478, (1952)</~~ref> independently proved the case ''r'' = 3 for all primes ''p''~~. ~~But in 1966 [[Guy Terjanian]] constructed a homogeneous polynomial of degree 4 over '''Q'''<sub>2<~~/~~sub> in 18 variables that has no non-trivial zero.<ref>Guy Terjanian, ''Un contre-exemple à une conjecture d'Artin'', C. R. Acad. Sci. Paris Sér. A–B, '''262''', A612, (1966)<~~/~~ref> On the other hand, the [[Ax–Kochen theorem]] shows that for any fixed degree Artin's conjecture is true for all but finitely many '''Q'''<sub>''p''<~~/~~sub>.~~

	~~== References ==~~
	~~{{reflist}}~~
	* {{cite book \| zbl=1125.11018 \| last=Davenport \| first=Harold \| authorlink=Harold Davenport \| title=Analytic methods for Diophantine equations and Diophantine inequalities \| others=Edited and prepared by T. D. Browning. With a preface by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman \| edition=2nd \| series=Cambridge Mathematical Library \| publisher=[[Cambridge University Press]] \| year=2005 \| isbn=0-521-60583-0 }}

	~~[[Category:Diophantine equations]]~~
	~~[[Category:Theorems in number theory]~~]

194.57.165.4: /* List of post hoc tests */

2013-11-04T17:51:32Z

List of post hoc tests

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en>MaynardClark: Included a Wikipedia page that works, since the page on Type I error rate had been deleted

2012-08-16T19:46:38Z

Included a Wikipedia page that works, since the page on Type I error rate had been deleted

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Post-hoc analysis - Revision history

en>LilHelpa: Typo fixing and general fixes using AWB

194.57.165.4: /* List of post hoc tests */

en>MaynardClark: Included a Wikipedia page that works, since the page on Type I error rate had been deleted