Minimal Polynomial is Irreducible
Theorem
Let $L/K$ be a field extension and $\alpha\in L$ 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 exists non-constant polynomials $g,h\in K[x]$ such that $f = gh$.
Applying the evaluation homomorphism to both sides of this equation at $\alpha$ we get the equality:
- $0 = f(\alpha) = g(\alpha)h(\alpha)$
in $K$.
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Since $K$ is a field, it is an integral domain and thus either $g(\alpha) = 0$ or $h(\alpha) = 0$ which contradicts the minimality of the degree of $f$.
$\blacksquare$
