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 such that for n ≥ ψ every equation
has a non-trivial solution in K.
Then, given homogeneous polynomials f1,...,fk of degrees r1,...,rk respectively with coefficients in K, for every set of positive integers r1,...,rk and every non-negative integer l, there exists a number ω such that for n ≥ ω there exists an l-dimensional affine subspace M of Kn 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, 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 r2. Indeed, Emil Artin conjectured that every homogeneous polynomial of degree r over Qp in more than r2 variables represents 0. This is obviously true for r = 1, and it is well known that the conjecture is true for r = 2. 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 Q2 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 Qp.