Fraction field of $F[x,y,z]/(x^4+y^4+zy)$ equal to $F(y,z)[x]/(x^4+y^4+zy)$?

Question

Why is the fraction field of $F[x,y,z]/(x^4+y^4+zy)$ equal to $F(y,z)[x]/(x^4+y^4+zy)$? Is there a general formula to compute the fraction field of $F[x_1,...,x_n]/(f)$ or $F[x_1,...,x_n]/(f_1,...,f_m)$

Answer 1

Hint:

This is because $\;F(y,z)\longrightarrow F(y,z)[x]/(x^4+y^4+zy)$ is an integral morphism and we have the following well-known result:

Let $B$ be an integral domain, $A$ a subring of $B$ such that $B$ is integral over $A$. Then $B$ is a field if and only if $A$ is a field.