Minimal Polynomial is Irreducible

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Theorem

Let $L / K$ be a field extension.

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


Then the minimal polynomial in $\alpha$ over $K$ is unique and irreducible.


Proof

Let $f \left({x}\right)$ be a minimal polynomial of $\alpha$ over $K$ of degree $n$.

From Minimal Polynomial is Unique we have that $f \left({x}\right)$ is unique.


Since $K$ is a field, we may assume that the coefficient of $x^n$ is $1$.

Aiming for a contradiction, 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 in $\alpha$:

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

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 $f$.

Hence the result, by Proof by Contradiction.

$\blacksquare$