Brauer's theorem

Brauer's theorem

:"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 fields in sufficiently many variables. [R. Brauer, "A note on systems of homogeneous algebraic equations", Bulletin of the American Mathematical Society, 51, pages 749-755 (1945)]

tatement 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

:(*)qquad a_1x_1^r+cdots+a_nx_n^r=0,quad a_iin K,quad i=1,ldots,n

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

:f_1(x_1,ldots,x_n)=cdots=f_k(x_1,ldots,x_n)=0,quadforall(x_1,ldots,x_n)in M.

An application to the field of p-adic numbers

Letting "K" be the field of p-adic numbers in the theorem, the equation (*) is satisfied, since mathbb{Q}_p^*/left(mathbb{Q}_p^* ight)^b, "b" a natural number, is finite. Choosing "k" = 1, one obtains the following corollary:

:A homogeneous equation "f"("x"1,...,"x""n") = 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"2. Indeed, Emil Artin conjectured ["Collected papers of Emil Artin", page x, Addison-Wesley, Reading, Mass., 1965] that every homogeneous polynomial of degree "r" over Q"p" in more than "r"2 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 [V. B. Demyanov, "On cubic forms over discrete normed fields",Dokl. Acad. Nauk SSSR 74(1950)pp 889-891] verfied the conjecture for "r"=3 and "p" eq3, and in 1952 D. J. Lewis [D. J. Lewis, "Cubic homogeneous polynomials over p-adic number fields", Annals of Mathematics, 56, pages 473-478, (1952)] independently proved the case "r"=3 for all primes "p". But in 1966 Guy Terjanian constructed a homogeneous polynomial of degree 4 over Q2 in 18 variables which doesn't have a non-trivial zero. [Guy Terjanian, "Un contre-example à une conjecture d'Artin", C. R. Acad. Sci. Paris Sér. A-B, 262, A612, (1966)] 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"p".

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Brauer–Suzuki theorem — In mathematics, the Brauer Suzuki theorem states that if a finite group has a generalized quaternion Sylow 2 subgroup and no non trivial normal subgroups of odd order, then the group has a centre of order 2. In particular, such a group cannot be… …   Wikipedia

  • Brauer-Siegel theorem — In mathematics, the Brauer Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the… …   Wikipedia

  • Brauer–Nesbitt theorem — In mathematics, the Brauer Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups. In modular representation theory,the Brauer Nesbitt theorem on blocks… …   Wikipedia

  • Brauer's theorem on induced characters — Brauer s theorem on induced characters, often known as Brauer s induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a …   Wikipedia

  • Alperin-Brauer-Gorenstein theorem — In mathematics, the Alperin Brauer Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2 subgroups. These are isomorphic either to three dimensional projective special linear or projective special… …   Wikipedia

  • Brauer's three main theorems — Brauer s main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic p ) with those of its p local subgroups, that is to say, the normalizers of its non trivial p subgroups.… …   Wikipedia

  • Brauer group — In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K . It is an abelian group with elements isomorphism classes of division algebras over K , such that the center is exactly K . The group is… …   Wikipedia

  • Richard Brauer — Infobox Scientist name = Richard Brauer box width = image width = 150px caption = Richard Brauer birth date = February 10, 1901 birth place = death date = April 17, 1977 death place = residence = citizenship = nationality = United States, Germany …   Wikipedia

  • Richard Brauer — Pour les articles homonymes, voir Brauer. Richard et Ilse Brauer en 1970 Richard Dagobert Brauer (10 février 1901 à Berlin – 17 avril 1977 à Belmont (Massachusetts)  …   Wikipédia en Français

  • Feit–Thompson theorem — In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963) Contents 1 History 2 Significance of the proof …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”