# Chevalley-Warning Theorem

Let ${\mathbb{F}}_{q}$ be the finite field^{} of $q$ elements with
characteristic^{} $p$. Let ${f}_{i}({x}_{1},\mathrm{\dots},{x}_{n})$, $i=1,2,\mathrm{\dots},r$,
be polynomial^{} of $n$ variables over ${\mathbb{F}}_{q}$. If $n>{\sum}_{i=1}^{r}\mathrm{deg}({f}_{i})$, then the number of solutions over
${\mathbb{F}}_{q}$ to the system of equations

${f}_{1}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ | $=0$ | ||

${f}_{2}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ | $=0$ | ||

$\mathrm{\vdots}$ | |||

${f}_{r}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ | $=0$ |

is divisible by $p$. In particular, if none of the polynomials ${f}_{1}$, ${f}_{2},\mathrm{\dots},{f}_{r}$ have constant term, then there are at least $p$ solutions.

Title | Chevalley-Warning Theorem |
---|---|

Canonical name | ChevalleyWarningTheorem |

Date of creation | 2013-03-22 17:46:52 |

Last modified on | 2013-03-22 17:46:52 |

Owner | kshum (5987) |

Last modified by | kshum (5987) |

Numerical id | 6 |

Author | kshum (5987) |

Entry type | Theorem |

Classification | msc 12E20 |