Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial

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Theorem

Let $L/K$ be a field extension.

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


Then there exists a unique irreducible monic polynomial $f\in K[x]$ such that $f(\alpha) = 0$, called the minimal polynomial.


Proof

Existence

Follows from Annihilating Polynomial of Minimal Degree is Irreducible.

$\Box$


Uniqueness

Let $f$ and $g$ be irreducible monic polynomials in $K[x]$ with $f(\alpha) = g(\alpha) = 0$.

Suppose $f$ and $g$ are distinct.

Then $f$ and $g$ are coprime.

Thus there exist polynomials $a, b \in k[x]$ with $af+bg = 1$.

Taking the evaluation homomorphism in $\alpha$, we obtain the contradiction that $0=1$.

$\blacksquare$