This should be very basic but I am having a bit of trouble finding the field of fractions for quotients of polynomial rings over a field. The specific example I am having trouble with is the following: Let $k$ be a field with characteristic $\neq 2$ and let $f \in k[x_1, x_2, \ldots , x_n]$ be a polynomial with no repeated factors. I am trying to determine the field of fractions of the ring $$ k[x_1, x_2, \ldots , x_n, z]/\langle z^2 - f\rangle. $$ The usual method I would use for a problem like this is the Chinese remainder theorem, but I don't see any way to apply that here. The other fact I know is that if $A$ is an integral domain with prime ideal $\mathfrak{p}$, then $\text{Frac}(A/\mathfrak{p}) \simeq A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. But the problem of finding $A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ seems no easier in this case. Can anyone point me in the right direction?

I am fairly certain I know what the result is, but I have no idea how to show it properly.

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    $\begingroup$ Isn't the field of fractions just $k(x_1, \ldots, x_n)[z]/(z^2 - f)$? It's certainly a field containing your ring, and i think you can argue that it is the smallest such. $\endgroup$ – André 3000 Feb 13 at 5:39
  • $\begingroup$ @André3000 that is the field of fractions, yes, but could you explain how exactly you'd show that? $\endgroup$ – Joe Feb 15 at 18:17

Let $R = k[x_1, x_2, \ldots , x_n, z]/(z^2 - f) = k[x_1, \ldots, x_n][\overline{z}]$, let $F = \operatorname{Frac}(R)$ be its field of fractions, and let $K = k(x_1, \ldots, x_n)[z]/(z^2 - f) = k(x_1, \ldots, x_n)[\overline{z}]$. Since $K$ is a field containing $R$ and $F$ is the smallest such field, then we have $R \subseteq F \subseteq K$. Now $F$ is a field containing $x_1, \ldots, x_n$, so it must also contain $k(x_1, \ldots, x_n)$ and since $F$ also contains $\overline{z}$ (the image of $z$ in the quotient), then $K \subseteq F$.


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