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In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.〔.〕〔.〕 The theorem is often given as this special case: If ''P'' is a polynomial function from C''n'' to C''n'' and ''P'' is injective then ''P'' is bijective. That is, if ''P'' always maps distinct arguments to distinct values, then the values of ''P'' cover all of C''n''.〔〔 The full theorem generalizes to any algebraic variety over an algebraically closed field.〔Éléments de géométrie algébrique, IV3, Proposition 10.4.11.〕 ==Proof via finite fields== Grothendieck's proof of the theorem〔〔 is based on proving the analogous theorem for finite fields and their algebraic closures. That is, for any field ''F'' that is itself finite or that is the closure of a finite field, if a polynomial ''P'' from ''Fn'' to itself is injective then it is bijective. If ''F'' is a finite field, then ''Fn'' is finite. In this case the theorem is true for trivial reasons having nothing to do with the representation of the function as a polynomial: any injection of a finite set to itself is a bijection. When ''F'' is the algebraic closure of a finite field, the result follows from Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over C would translate into a counterexample in some algebraic extension of a finite field. This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic 0 translate into algebraic relations over finite fields with large characteristic.〔 Thus, one can use the arithmetic of finite fields to prove a statement about C even though there is no homomorphism from any finite field to C. The proof thus uses model theoretic principles to prove an elementary statement about polynomials. The proof for the general case uses a similar method. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ax–Grothendieck theorem」の詳細全文を読む スポンサード リンク
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