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Math Help - Quotient of polynomial ring is a localization of a polynomial ring

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    Quotient of polynomial ring is a localization of a polynomial ring

    Let p(x,y,z)=z^3-y(y^2-x^2)(x-1)\in k[x,y,z], where k is an algebraically closed field. The goal is to show that some ring obtained by inverting certain elements of T=k[x,y,z]/(p) is isomorphic to a polynomial ring, again with certain elements inverted. This will show that the quotient field of T is isomorphic to the field of rational functions in some (should be 2) variables (which is the ACTUAL goal).

    My first thought was to consider the ring T[\frac{1}{y}], where the relationship  z^3=y(y^2-x^2)(x-1) can be rewritten as (\frac{z}{y})^3=(1-(\frac{x}{y})^2)(x-1). Then do a change of variables, say w=z/x,v=y/x, and then solve for x. That relation should then let me define a map to some localization of k[v,w].

    I could be doing it right, but the equations seem to get way too complicated to check whether or not I have an isomorphism. Anyone have an idea?
    Last edited by topspin1617; October 22nd 2011 at 08:25 PM.
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    Re: Quotient of polynomial ring is a localization of a polynomial ring

    Quote Originally Posted by topspin1617 View Post
    Let p(x,y,z)=z^3-y(y^2-x^2)(x-1)\in k[x,y,z], where k is an algebraically closed field. The goal is to show that some ring obtained by inverting certain elements of T=k[x,y,z]/(p) is isomorphic to a polynomial ring, again with certain elements inverted. This will show that the quotient field of T is isomorphic to the field of rational functions in some (should be 2) variables (which is the ACTUAL goal).

    My first thought was to consider the ring T[\frac{1}{y}], where the relationship  z^3=y(y^2-x^2)(x-1) can be rewritten as (\frac{z}{y})^3=(1-(\frac{x}{y})^2)(x-1). Then do a change of variables, say w=z/x,v=y/x, and then solve for x. That relation should then let me define a map to some localization of k[v,w].

    I could be doing it right, but the equations seem to get way too complicated to check whether or not I have an isomorphism. Anyone have an idea?
    let S = k[x,y,z], where x,y are algebraically independent and z^3=y(y^2-x^2)(x-1). clearly T \cong S. let u = \frac{y}{x} and v = \frac{z}{x}. then x = 1 + \frac{v^3}{u(u^2-1)} and thus the quotient field of S is Q(S) = k(u,v).
    so the only thing you need to prove is that u and v are algebraically independent.
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