Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is constant. Does this help me somehow?
As I said below, it seems that it is enough to distinguish $\mathrm{Frac} \ \mathbf{C}[x,y]/(x^3+y^3-1)$ and $\mathbf{C}(x)$, how can we do this?