Chevalley–Warning theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields . Statement of the theorems. Let formula_1 be a finite field and formula_2 be a set of polynomials such that the number of variables satisfies where formula_4 is the total degree of formula_5. The theorems are statements about  the solutions of the following system of polynomial equations Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since formula_8 is at least 2. Both theorems are best possible in the sense that, given any formula_16, the list formula_17 has total degree formula_16 and only the trivial solution. Alternatively, using just one polynomial, we can take "f" to be the degree "n" polynomial given by the norm of "x""a" + ... + "x""a" where the elements "a" form a basis of the finite field of order "p". Warning proved another theorem, known as Warning's second theorem, which  states that if the system of polynomial equations has the trivial solution, then it has at least formula_19 solutions where formula_20 is the size of the finite field and formula_21. Chevalley's theorem also follows directly from this. Proof of Warning's theorem. "Remark:" If formula_22 then so the sum over formula_24 of any polynomial in formula_25 of degree less than formula_26 also vanishes. The total number of common solutions modulo formula_8 of formula_28 is equal to because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials formula_30 is less  than "n" then this vanishes by the remark above. Artin's conjecture. It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem. The Ax–Katz theorem. The Ax–Katz theorem, named after James Ax and Nicholas Katz, determines more accurately a power formula_31 of the cardinality formula_20 of formula_1 dividing the number of  solutions; here, if formula_34 is the largest of the formula_4, then the exponent formula_36 can be taken as the ceiling function of The Ax–Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. Namely, the same power of formula_20 divides each of these algebraic integers.