Brauer's theorem on forms
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. 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 has a non-trivial (i.e. not all "x" are equal to 0) solution in "K". Then, given homogeneous polynomials "f"...,"f" of degrees "r"...,"r" respectively with coefficients in "K", for every set of positive integers "r"...,"r  and every non-negative integer l", there exists a number ω("r"...,"r","l") such that for "n" ≥ ω("r"...,"r","l") there exists an "l"-dimensional affine subspace "M" of "K" (regarded as a vector space over "K") satisfying 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 formula_3, "b" a natural number, is finite. Choosing "k" = 1, one obtains the following corollary: One can show that if "n" is sufficiently large according to the above corollary, then "n" is greater than "r". Indeed, Emil  Artin conjectured that every homogeneous polynomial of degree "r" over Q in more than "r" 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 verified the conjecture for "r" = 3 and "p" ≠ 3, and in 1952 D. J. Lewis independently proved the case "r" = 3 for all primes "p". But in 1966 Guy Terjanian constructed a homogeneous polynomial  of degree 4 over Q in 18 variables that has no non-trivial zero. 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.