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Given a field of the form $\mathbb C(t)[g]$, where $t$ is transcendental over $\mathbb C$ and $g$ is algebraic over $\mathbb C(t)$ do I always find an irreducible polynomial $F\in \mathbb C[X,Y]$ such that $Quot(\mathbb C[X,Y]/(F))\cong \mathbb C(t)[g]$?

I thought about taking the minimal polynomial of $g$ over $\mathbb C(t)$ and killing the denominators, but I couldn't manage writing a serious proof.

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    I know the primitive element theorem I don't see, why it should give me a solution. Beeing a little bit more precise would be helpful. Thanks.2012-12-27
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    I misread the problem statement. You do not need the primitive element theorem for the body question (but the fact that the body and title question are equivalent uses the primitive element theorem). You are correct that you should use the minimal polynomial of $g$ over $\mathbb{C}(f)$ and then clear denominators.2012-12-28
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    @beginner I've changed $f$ by $t$ suggesting "transcendental".2012-12-29

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