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Let $C$ be a curve in $\mathbb{P}^2$ over a field $K$ defined by an equation of the form $f: y^2 - x^3 - ax^2$ with $a \in K$.

I want to show that the function field of $C$ is isomorphic to $K(t)$.

The function field $K(C)$ is the fraction field of $K[C] = K[x,y]/(f)$.

Since we're quotienting by $f$, I think that every $g \in K(C)$ can be written as $a(x) + b(x)y$, where $a,b \in K(x)$. Therefore $K(C) \cong K(x)[y]/(f)$. (This coincides so far with a comment on a similar question here.)

However, I don't know how to get the last part. Is there an obvious isomorphism I'm overlooking, or do I have to use the equation of the curve somewhere explicitly? (I've shown that there exists a singular point on $C$, but I don't see any use of this fact.)

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    $\begingroup$ Try $t=y/x$. From $f=0$ you can solve $t^2=x+a$. From there on it should be easy. $\endgroup$
    – Jyrki Lahtonen
    Commented May 31, 2020 at 18:35
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    $\begingroup$ Basically, the curve has a double point at the origin, and this is the usual blow-up resolving the singularity. $\endgroup$
    – Jyrki Lahtonen
    Commented May 31, 2020 at 18:41
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    $\begingroup$ A geometric interpretation of Jyrki Lahtonen's idea is that this gives a rational parameterization of your curve, which shows that your curve is birational to a line and thus has function field $K(t)$. $\endgroup$
    – KReiser
    Commented May 31, 2020 at 19:27

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