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When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 correspondence with the functions defined by these fractions? Or might there be two different fractions (thus not fractions with one of them not reduced or something) which define the same function in $x1, x2, ..., xn$ ??

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