Two field extensions of transcendence degree 1

Question

Let $K$ be a field, $K_1:=FRAC(K[x_1,\dots,x_n]/P)$ and $K_2:=FRAC(K[y_1,\dots,y_m]/Q)$ with prime ideals $P$ and $Q$, such that there is a field injection $\varphi:K_1\to K_2$. Assuming that the transcendence degree of $K_1/K$ and $K_2/K$ are $1$ I want to show that $[K_1:K_2]<\infty$.

Much thanks to everybody taking time to think about this.

Best regards.

Answer 1

By your assumption $K_2$ is algebraic over $K_1$ (since they have the same finite transcendence degree over a common subfield) and furthermore it is finitely generated over $K_1$ (since it is already finitely generated over $K$). It is well known and trivial that finitely generated + algebraic = finite.