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Math 2.0

Number theory Diophantine equations Theorem

Chevalley–Warning theorem

Statement

Let $\mathbb{F}$ be a finite field of characteristic $p$ . Let $f_1, \dots, f_r \in \mathbb{F}[X_1, \dots, X_n]$ be polynomials with total degrees $d_1, \dots, d_r$ , such that $n > d_1 + \dots + d_r$ . Then the number of common zeros $(a_1, \dots, a_n) \in \mathbb{F}^n$ of $f_1, \dots, f_r$ is divisible by $p$ .

Remarks

A corollary is Chevalley's theorem: if the polynomials have no constant terms, then they have a common zero other than $(0, \dots, 0)$ . The reason is that $(0, \dots, 0)$ is a zero, and the number of zeros is a multiple of $p \ge 2$ . For example, a quadratic form in $3$ or more variables over $\mathbb{Z}/p\mathbb{Z}$ has a nontrivial zero.

History

Ewald Warning and Claude Chevalley proved the results in 1935.

Where this leads

2 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem, Statement of the theorems

About this page

Tag	`08895Q5W96`
URI	`m2://08895Q5W96`
Kind	statement, theorem
Writing	W0 stub
MSC	11T06
Status	This copy of the site has no build data. Run `m2 status 08895Q5W96`.
Same as	Wikidata Q720469 Wikipedia Chevalley–Warning theorem

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Quasi-algebraically closed field