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High Quality Content by WIKIPEDIA articles!In mathematics, the Ax Grothendieck theorem is a result that was proved independently by James Ax and Alexander Grothendieck 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, and true more generally for arbitrary functions, not just for polynomials: any injection of a finite set to itself is a bijection.