Let $R=\mathbb{C}[x,y,z,t]/(xz-y^{2},yt-z^{2},xt-yz)$ (i have proved in a previous question that is an integral domain).
I am trying to calculate the transcendence degree of $Frac(R)$ over $\mathbb{C}$.
This what i tried:
Consider the extensions $\mathbb{C} \subset Frac(R) \subset C(u,v)$ , since $trdeg_{\mathbb{C}}\mathbb{C}(u,v)=2<\infty$ , we know that $trdeg_{\mathbb{C}}C(u,v)=trdeg_{\mathbb{C}}Frac(R)+trdeg_{Frac(R)}\mathbb{C}(u,v)$.
Do you have any idea about proving $trdeg_{Frac(R)}\mathbb{C}(u,v)=0$? It seems rather difficult to compute $Frac(R)$ and dont really see any other way to show the result. Thank you in advance!