Minimal Polynomial is Irreducible

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

Let $\alpha \in L$ be algebraic over $K$.

Then the minimal polynomial of $\alpha$ over $K$ is irreducible.

Proof
Let $f$ be a minimal polynomial of $\alpha$ over $K$.

Suppose that $f$ is not irreducible.

Then there exist non-constant polynomials $g, h \in K \left[{x}\right]$ such that $f = g h$.

By definition of $f$ as the minimal polynomial of $\alpha$:


 * $0 = f \left({\alpha}\right) = g \left({\alpha}\right) h \left({\alpha}\right)$

Since $L$ is a field, it is an integral domain.

Therefore, as $g \left({\alpha}\right), h \left({\alpha}\right) \in L$, either $g \left({\alpha}\right) = 0$ or $h \left({\alpha}\right) = 0$.

This contradicts the minimality of the degree of $f$.