Theorem 1   Let $L/K$ be a finite field extension. Then $L/K$ is an algebraic extension.

Proof. In order to prove that $L/K$ is an algebraic extension, we need to show that any element $\alpha\in L$ is algebraic, i.e., there exists a non-zero polynomial $p(x)\in K[x]$ such that $p(\alpha)=0$ .

Recall that $L/K$ is a finite extension of fields, by definition, it means that $L$ is a finite dimensional vector space over $K$ . Let the dimension be $$[L\colon K]=n$$ for some $n\in \Nats$ .

Consider the following set of ``vectors'' in $L$ : $$\mathcal{S}=\{ 1, \alpha, \alpha^2,\alpha^3,\ldots,\alpha^n\}$$ Note that the cardinality of $S$ is $n+1$ , one more than the dimension of the vector space. Therefore, the elements of $S$ must be linearly dependent over $K$ , otherwise the dimension of $S$ would be greater than $n$ . Hence, there exist $k_i\in K,\ 0\leq i \leq n$ , not all zero, such that $$k_0+k_1\alpha+k_2\alpha^2+k_3\alpha^3+\ldots+k_n\alpha^n=0$$ Thus, if we define $$p(X)=k_0+k_1X+k_2X^2+k_3X^3+\ldots+k_nX^n$$ then $p(X)\in K[X]$ and $p(\alpha)=0$ , as desired.