I'm looking for a proof for this proposition: "Let $L/K$ be a field extension, where $L$ is a finitely generated $K$ algebra. Then $L/K$ is a finite field extension."
My attempt: In "Field and Galois Theory" written by Morandi, there is a proposition saying that a field extension $K/F$ is finite if and only if $K$ is finitely generated and algebraic over $F$.
Let $K[a_1,...,a_n]/I\cong L$ for some $a_i\in L$ and $I$ is an ideal of $L$. I tried to show that $\{a_i+I\}$ are algebraic over $K$
$L=K[a_1,\ldots,a_n]$ is a field.
If $L/K$ is algebraic then $L$ is a quotient of $K[x_1,\ldots,x_n]/(f_1(x_1),\ldots,f_n(x_n))$ which is a $\prod_j \deg(f_j)$ vector space so $L$ is a finite dimensional vector space (where $f_j\in K[x]$ is the minimal polynomial of $a_j$).
If $L/K$ is not algebraic then take a transcendental basis $a_{i_1},\ldots,a_{i_m}$.
Let $t=a_{i_m}$, $F=K(a_{i_1},\ldots,a_{i_{m-1}})$,
$L$ is an algebraic extension of $F(t)$, it is a quotient of $F(t)[x_1,\ldots,x_n]/(f_1(x_1),\ldots,f_n(x_n))$ where $f_j$ is now the monic $F(t)$-minimal polynomial of $a_j$. The coefficients of $f_j$ are $u_{j,l}(t)/v_{j,l}(t)\in Frac(F[t])$, let $$h(t)= \prod_{j,l} v_{j,l}(t)\in F[t]$$ Then $$L= F[t,h(t)^{-1}][a_1,\ldots,a_n]$$ is integral over $F[t,h(t)^{-1}]$. Take an irreducible polynomial $g(t)\in F[t]$ not dividing $h(t)$,
$g(t)^{-1}$ is not integral over $F[t,h(t)^{-1}]$ so $g(t)^{-1}\not \in L$ which is a contradiction.