In algebra, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).
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Statement of the theorems
Let
where
Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since
Both theorems are best possible in the sense that, given any
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
Proof of Warning's theorem
Remark: If
so the sum over
The total number of common solutions modulo
because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials
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
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