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.
[1] 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 xi are equal to 0) 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(r1,...,rk,l) such that for n ≥ ω(r1,...,rk,l) there exists an l-dimensional affine subspace M of Kn (regarded as a vector space over K) satisfying Letting K be the field of p-adic numbers in the theorem, the equation (*) is satisfied, since
Indeed, Emil Artin conjectured[2] that every homogeneous polynomial of degree r over Qp in more than r2 variables represents 0.
In 1950 Demyanov[3] verified the conjecture for r = 3 and p ≠ 3, and in 1952 D. J. Lewis[4] 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.
[5] 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.