Showing the fractional field of $\Bbb C[x,y]/(y^2-x^3+x)$ is not isomorphic to $\Bbb C(t)$

Question

Consider the integral domain $R=\Bbb C[x,y]/(y^2-x^3+x)$. How can we show that the fractional field $K$ of $R$ is not isomorphic to $\Bbb C(t)$?

I attempted as follows: Suppose $K$ is isomorphic to $\Bbb C(t)$. Under this isomorphism let $x$ and $y$ correspond to $f(t)/g(t)$ and $h(t)/k(t)$, respectively, where $\gcd(f,g)=1=\gcd(h,k)$. Then from $y^2=x^3-x$ we get $f^2h^3=g^2k(k^2-h^2)$, so $k$ divides $f$. Maybe using this I can get a contradiction, but I got stuck here.

Answer 1

The field of fractions of $R$ is $K=\mathbb C(x)[\sqrt{x^3-x}]$, or $K=\mathbb C(x,\sqrt{x^3-x})$.

Now notice that $K$ is not a purely transcendental extension of $\mathbb C$; see Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?.