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Math Help - irreducibility of a algebraic set

  1. #1
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    irreducibility of a algebraic set

    Show that the image of \phi : K \rightarrow K^3 given by \phi(t) = (t,f(t),g(t)) with f,g \in K[x] (an algebraic set) is irreducible.

    thanks
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  2. #2
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    Quote Originally Posted by roporte View Post
    Show that the image of \phi : K \rightarrow K^3 given by \phi(t) = (t,f(t),g(t)) with f,g \in K[x] (an algebraic set) is irreducible.

    thanks
    an algebraic set X is irreducible (or affine variety) iff \mathcal{I}(X) is prime. so from this thread we only need to show that the ideal J=<y-f(x), z - g(x)> is a prime ideal of  K[x,y,z].

    to see this define the map \varphi: K[x,y,z] \longrightarrow K[x] by: \varphi(h(x,y,z))=h(x,f(x),g(x)). since \varphi is just a simple evaluation, it's a ring homomorphiam. also \varphi is onto since \varphi(h(x))=h(x),

    for any h(x) \in K[x]. finally observe that \ker \varphi = J. thus: \frac{K[x,y,z]}{J} \simeq K[x]. now since K[x] is an integral domain, J must be prime. Q.E.D.
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