On Rational function fields of varieties

Question

1. It is given that $Q=V(x^3-yz^2)$ is a variety in $\mathbb P^2$ (i.e. projective space of dimension two).Then what is the rational function field K(Q) of Q?

2.Consider the hypersurface $x^2+y^2+z^2+w^2=0$ in $\mathbb P^3$.Then what is the field of rational functions for this hypersurface?

I am getting hard to determine the rational function fields and also dont know the exact answer.Any help is appreciated.

Answer 1

For 1., we can look at an affine chart of $Q$. In the chart $z=1$, the variety is given by $x^3=y$, which is smooth of dimension one.

Now $K(Q)=K(U)$ for any dense open set of $Q$, so...

For 2, again set $x=1$. Then $y=i\sqrt{1+z^2+w^2}$.