Chevalley–Warning theorem
# 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

    $f_{j}(x_{1},\dots ,x_{n})=0\quad {\text{for}}\,j=1,\ldots ,r.$
* The 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 $p$.
* The Chevalley theorem states that if the system has the trivial solution $(0,\dots ,0)\in \mathbb {F} ^{n}$, that is, 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 $f_1$ to be the degree n polynomial given by the norm of $x_1a_1$ + ... + $x_na_n$ where the elements a form a basis of the finite field of order $p^n.$

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 $i<q-1$ 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.

