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| In algebra, the '''Chevalley–Warning theorem''' implies that certain [[polynomial|polynomial equations]] in sufficiently many variables over a [[finite field]] have solutions. It was proved by {{harvs|txt|first=Ewald |last=Warning|year=1936}} and a slightly weaker form of the theorem, known as '''Chevalley's theorem''', was proved by {{harvs|txt|authorlink=Claude Chevalley|last=Chevalley|year= 1936}}. Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are [[quasi-algebraically closed field]]s {{harv|Artin|1982|loc=page x}}.
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| == Statement of the theorems ==
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| Let <math>\mathbb{F}</math> be a finite field and <math>\{f_j\}_{j=1}^r\subseteq\mathbb{F}[X_1,\ldots,X_n]</math> be a set of polynomials such that the number of variables satisfies
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| :<math>n>\sum_{j=1}^r d_j</math> | |
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| where <math>d_j</math> is the [[total degree]] of <math>f_j</math>. The theorems are statements about the solutions of the following system of polynomial equations
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| :<math>f_j(x_1,\dots,x_n)=0\quad\text{for}\, j=1,\ldots, r.</math>
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| * ''Chevalley–Warning theorem'' states that the number of common solutions <math>(a_1,\dots,a_n) \in \mathbb{F}^n</math> is divisible by the [[characteristic (algebra)|characteristic]] <math>p</math> of <math>\mathbb{F}</math>. Or in other words, the cardinality of the vanishing set of <math>\{f_j\}_{j=1}^r</math> is <math>0</math> modulo <math>p</math>.
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| * ''Chevalley's theorem'' states that if the system has the trivial solution <math>(0,\dots,0) \in \mathbb{F}^n</math>, i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution <math>(a_1,\dots,a_n) \in \mathbb{F}^n \backslash \{(0,\dots,0)\}</math>.
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| Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since <math>p</math> is at least 2.
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| Both theorems are best possible in the sense that, given any <math>n</math>, the list <math>f_j = x_j, j=1,\dots,n</math> has total degree <math>n</math> and only the trivial solution. Alternatively, using just one polynomial, we can take ''f''<sub>1</sub> to be the degree ''n'' polynomial given by the norm of ''x''<sub>1</sub>''a''<sub>1</sub> + ... + ''x''<sub>''n''</sub>''a''<sub>''n''</sub> where the elements ''a'' form a basis of the finite field of order ''p''<sup>''n''</sup>.
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| ==Proof of Warning's theorem==
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| ''Remark:'' If <math>i<p-1</math> then
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| :<math>\sum_{x\in\mathbb{F}}x^i=0</math>
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| so the sum over <math>\mathbb{F}^n</math> of any polynomial in <math>x_1,\ldots,x_n</math> of degree less than <math>n(p-1)</math> also vanishes.
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| The total number of common solutions modulo <math>p</math> of <math>f_1, \ldots, f_r = 0</math> is equal to
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| :<math>\sum_{x\in\mathbb{F}^n}(1-f_1^{p-1}(x))\cdot\ldots\cdot(1-f_r^{p-1}(x)) </math>
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| because each term is 1 for a solution and 0 otherwise. | |
| If the sum of the degrees of the polynomials <math>f_i</math> is less than ''n'' then this vanishes by the remark above.
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| == Artin's conjecture ==
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| 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 little theorem|Wedderburn's theorem]].
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| == The Ax–Katz theorem ==
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| The '''Ax–Katz theorem''', named after [[James Ax]] and [[Nicholas Katz]], determines more accurately a power <math>q^b</math> of the cardinality <math>q</math> of <math>\mathbb{F}</math> dividing the number of solutions; here, if <math>d</math> is the largest of the <math>d_j</math>, then the exponent <math>b</math> can be taken as the [[ceiling function]] of
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| : <math>\frac{n - \sum_j d_j}{d}.</math>
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| 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 <math>q</math> divides each of these [[algebraic integer]]s.
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| == See also ==
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| * [[combinatorial Nullstellensatz]]
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| ==References==
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| *{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | editor1-last=Lang | editor1-first=Serge. | editor2-last=Tate | editor2-first=John | editor2-link=John Tate | title=Collected papers | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90686-7 | mr=671416 | year=1982}}
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| *{{citation| last=Ax | first=James | authorlink=James Ax | year=1964 | title=Zeros of polynomials over finite fields | journal=American Journal of Mathematics | volume=86 | pages=255–261 | mr=0160775 | doi=10.2307/2373163}}
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| *{{citation| last=Chevalley | first=Claude | year=1936 | title=Démonstration d'une hypothèse de M. Artin | language=French| journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=73–75 | zbl=0011.14504 |jfm=61.1043.01| doi=10.1007/BF02940714}}
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| *{{citation| last=Katz | first=Nicholas M. | authorlink=Nicholas Katz | year=1971 | title=On a theorem of Ax | journal=Amer. J. Math. | volume=93 | issue=2 | pages=485–499 | doi=10.2307/2373389}}
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| *{{citation| last=Warning | first=Ewald | year=1936 | title=Bemerkung zur vorstehenden Arbeit von Herrn Chevalley|language=German | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=76–83 | zbl=0011.14601 |jfm=61.1043.02| doi=10.1007/BF02940715}}
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| *{{citation| last=Serre | first=Jean-Pierre | year=1973 | title=A course in arithmetic| pages=5–6 | isbn=0-387-90040-3}}
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| {{DEFAULTSORT:Chevalley-Warning theorem}}
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| [[Category:Finite fields]]
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| [[Category:Diophantine geometry]]
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| [[Category:Theorems in algebra]]
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