View source for Brauer's theorem on forms

{{short description|On the representability of 0 by forms over certain fields in sufficiently many variables}}
:''There also is [[Brauer's theorem on induced characters]].''

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&nbsp;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''&nbsp;=&nbsp;1, and it is well known that the conjecture is true for ''r''&nbsp;=&nbsp;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''&nbsp;=&nbsp;3 and ''p''&nbsp;≠&nbsp;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''&nbsp;=&nbsp;3 for all primes&nbsp;''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>.

== Notes ==

* {{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 }}

== References ==
{{reflist}}

[[Category:Diophantine equations]]
[[Category:Theorems in number theory]]
[[Category:P-adic numbers]]