Finite Field Extension is Algebraic

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Theorem

Let $L / K$ be a finite field extension.

Then $L / K$ is algebraic.

Proof

Let $x \in L$ be arbitrary.

Let $n = \index L K$ be the degree of $L$ over $K$.

From Size of Linearly Independent Subset is at Most Size of Finite Generator, there is a $K$-linear combination of $\set {1, \ldots, x^n}$ equal to $0$.

Say $a_n x^n + \cdots + a_1 x + a_0 = 0$, $a_i \in K$, $i = 0, \ldots, n$.

Therefore $x$ satisfies a polynomial with coefficients in $K$.

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That is, $x$ is algebraic.

Since $x \in L$ was chosen arbitrarily, $L / K$ is algebraic.

$\blacksquare$