## Finite extension is Algebraic extension (Proof) + “Converse”

These two are useful lemmas in Galois/Field Theory.

Finite extension is Algebraic extension (Proof)
Let $L/K$ be a finite field extension. Then $L/K$ is an algebraic extension.
Proof:
Let $L/K$ be a finite extension, where $[L:K]=n$. Let $\alpha\in L$. Consider $\{1,\alpha,\alpha^2,\dots,\alpha^n\}$ which has to be linearly dependent over $K$ since there are $n+1$ elements. Thus, there exists $c_i\in K$ (not all zero) such that $\sum_{i=0}^n c_i\alpha^i=0$, so $\alpha$ is algebraic over $K$.

Finitely Generated Algebraic Extension is Finite (Proof)
Let $L/K$ be a finitely generated algebraic extension. Then $L/K$ is a finite extension.

Proof:
Since $L/K$ is finitely generated, $L=K(\alpha_1,\dots,\alpha_n)$ for some $\alpha_1,\dots,\alpha_n\in K$. Since $L/K$ is algebraic, each $\alpha_i$ is algebraic over $K$. Denote $L_i:=K(\alpha_1,\dots,\alpha_i)$ for $1\leq i\leq n$. Then $L_i=L_{i-1}(\alpha_i)$ for each $i$. Since $\alpha_i$ is algebraic over $K$, it is also algebraic over $L_{i-1}$, so there exists a polynomial $g_i$ with coefficients in $L_{i-1}$ such that $g_i(\alpha_i)=0$. Thus $[L_i:L_{i-1}]\leq\deg g_i<\infty$. Similarly $[L_1:K]<\infty$. By Tower Law, $[L:K]=[L_n:L_{n-1}][L_{n-1}:L_{n-2}]\dots[L_1:K]<\infty$.