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 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).

Statement of the theorems

Let $\mathbb {F}$ be a finite field and $\{f_{j}\}_{j=1}^{r}\subseteq \mathbb {F} [X_{1},\ldots ,X_{n}]$ be a set of polynomials such that the number of variables satisfies

$n>\sum _{j=1}^{r}d_{j}$ where $d_{j}$ is the total degree of $f_{j}$ . The theorems are statements about the solutions of the following system of polynomial equations

$f_{j}(x_{1},\dots ,x_{n})=0\quad {\text{for}}\,j=1,\ldots ,r.$ • Chevalley–Warning theorem states that the number of common solutions $(a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}$ is divisible by the characteristic $p$ of $\mathbb {F}$ . Or in other words, the cardinality of the vanishing set of $\{f_{j}\}_{j=1}^{r}$ is $0$ modulo $p$ .
• Chevalley's theorem states that if the system has the trivial solution $(0,\dots ,0)\in \mathbb {F} ^{n}$ , i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution $(a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}\backslash \{(0,\dots ,0)\}$ .

Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since $p$ is at least 2.

Both theorems are best possible in the sense that, given any $n$ , the list $f_{j}=x_{j},j=1,\dots ,n$ has total degree $n$ and only the trivial solution. Alternatively, using just one polynomial, we can take f1 to be the degree n polynomial given by the norm of x1a1 + ... + xnan where the elements a form a basis of the finite field of order pn.

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 $q^{n-d}$ solutions where $q$ is the size of the finite field and $d:=d_{1}+\dots +d_{r}$ . Chevalley's theorem also follows directly from this.

Proof of Warning's theorem

Remark: If $i then

$\sum _{x\in \mathbb {F} }x^{i}=0$ so the sum over $\mathbb {F} ^{n}$ of any polynomial in $x_{1},\ldots ,x_{n}$ of degree less than $n(q-1)$ also vanishes.

The total number of common solutions modulo $p$ of $f_{1},\ldots ,f_{r}=0$ is equal to

$\sum _{x\in \mathbb {F} ^{n}}(1-f_{1}^{q-1}(x))\cdot \ldots \cdot (1-f_{r}^{q-1}(x))$ because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials $f_{i}$ 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 $q^{b}$ of the cardinality $q$ of $\mathbb {F}$ dividing the number of solutions; here, if $d$ is the largest of the $d_{j}$ , then the exponent $b$ can be taken as the ceiling function of

${\frac {n-\sum _{j}d_{j}}{d}}.$ 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 $q$ divides each of these algebraic integers.