Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial
Let $L/K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$.
Follows from Annihilating Polynomial of Minimal Degree is Irreducible.
Suppose $f$ and $g$ are distinct.
Then $f$ and $g$ are coprime.
Thus there exist polynomials $a, b \in k[x]$ with $af+bg = 1$.