In number theory, theChevalley–Warning theorem implies that certainpolynomial equations in sufficiently many variables over afinite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known asChevalley's theorem, was proved byChevalley (1935). Chevalley's theorem impliedArtin's andDickson's conjecture that finite fields arequasi-algebraically closed fields (Artin 1982, page x).

Let${\displaystyle \mathbb{F}}$ be a finite field and${\displaystyle \{f_j{\}}_{j=1}^r\subseteq \mathbb{F}[X_1,\dots ,X_n]}$ be a set of polynomials such that the number of variables satisfies

${\displaystyle n>\sum _{j=1}^r{d}_j}$

where${\displaystyle d_j}$ is thetotal degree of${\displaystyle f_j}$. The theorems are statements about the solutions of the following system of polynomial equations

${\displaystyle f_j(x_1,\dots ,x_n)=0\text{for}j=1,\dots ,r.}$

• TheChevalley–Warning theorem states that the number of common solutions${\displaystyle (a_1,\dots ,a_n)\in {\mathbb{F}}^n}$ is divisible by thecharacteristic${\displaystyle p}$ of${\displaystyle \mathbb{F}}$. Or in other words, the cardinality of the vanishing set of${\displaystyle \{f_j{\}}_{j=1}^r}$ is${\displaystyle 0}$ modulo${\displaystyle p}$.
• TheChevalley theorem states that if the system has the trivial solution${\displaystyle (0,\dots ,0)\in {\mathbb{F}}^n}$, that is, if the polynomials have no constant terms, then the system also has a non-trivial solution${\displaystyle (a_1,\dots ,a_n)\in {\mathbb{F}}^n\mathrm{\setminus }\{(0,\dots ,0)\}}$.

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

Both theorems are best possible in the sense that, given any${\displaystyle n}$, the list${\displaystyle f_j=x_j,j=1,\dots ,n}$ has total degree${\displaystyle n}$ and only the trivial solution. Alternatively, using just one polynomial, we can takef₁ to be the degreen polynomial given by thenorm ofx₁a₁ + ... +x_na_n where the elementsa form a basis of the finite field of orderpⁿ.

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${\displaystyle q^{n-d}}$ solutions where${\displaystyle q}$ is the size of the finite field and${\displaystyle d:=d_1+\cdots +d_r}$. Chevalley's theorem also follows directly from this.

Remark:^[1] If then

${\displaystyle \sum _{x\in \mathbb{F}}{x}^i=0}$

so the sum over${\displaystyle {\mathbb{F}}^n}$ of any polynomial in${\displaystyle x_1,\dots ,x_n}$ of degree less than${\displaystyle n(q-1)}$ also vanishes.

The total number of common solutions modulo${\displaystyle p}$ of${\displaystyle f_1,\dots ,f_r=0}$ is equal to

${\displaystyle \sum _{x\in {\mathbb{F}}^n}(1-f_1^{q-1}(x))\cdot \dots \cdot (1-f_r^{q-1}(x))}$

because each term is 1 for a solution and 0 otherwise.If the sum of the degrees of the polynomials${\displaystyle f_i}$ is less thann then this vanishes by the remark above.

It is a consequence of Chevalley's theorem that finite fields arequasi-algebraically closed. This had been conjectured byEmil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivialBrauer group, together with the fact that finite fields have trivial Brauer group byWedderburn's theorem.

TheAx–Katz theorem, named afterJames Ax andNicholas Katz, determines more accurately a power${\displaystyle q^b}$ of the cardinality${\displaystyle q}$ of${\displaystyle \mathbb{F}}$ dividing the number of solutions; here, if${\displaystyle d}$ is the largest of the${\displaystyle d_j}$, then the exponent${\displaystyle b}$ can be taken as theceiling function of

${\displaystyle \frac{n-\sum _j{d}_j}{d}.}$

The Ax–Katz result has an interpretation inétale cohomology as a divisibility result for the (reciprocals of) the zeroes and poles of thelocal zeta-function. Namely, the same power of${\displaystyle q}$ divides each of thesealgebraic integers.

• Combinatorial Nullstellensatz

• Artin, Emil (1982), Lang, Serge.;Tate, John (eds.),Collected papers, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90686-7,MR 0671416
• Ax, James (1964), "Zeros of polynomials over finite fields",American Journal of Mathematics,86:255–261,doi:10.2307/2373163,MR 0160775
• Chevalley, Claude (1935), "Démonstration d'une hypothèse de M. Artin",Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in French),11:73–75,doi:10.1007/BF02940714,JFM 61.1043.01,Zbl 0011.14504
• Katz, Nicholas M. (1971), "On a theorem of Ax",Amer. J. Math.,93 (2):485–499,doi:10.2307/2373389
• Warning, Ewald (1935), "Bemerkung zur vorstehenden Arbeit von Herrn Chevalley",Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German),11:76–83,doi:10.1007/BF02940715,JFM 61.1043.02,Zbl 0011.14601
• Serre, Jean-Pierre (1973),A course in arithmetic, pp. 5–6,ISBN 0-387-90040-3

• "Proofs of the Chevalley-Warning theorem".

• Finite fields
• Diophantine geometry
• Theorems in algebra

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