Let $K$ be an extension field of $F$, and let $\alpha \in K$ be algebraic. Suppose that $f(x) \in F[x]$. Prove that $f(x)$ is the unique monic polynomial of least degree with f($\alpha$) = $0$.
My thinking is that if $f(x)$ isn't unique, then there exists another polynomial of $g(x)$ of the same degree. And, if that's the case, then these two polynomials would divide each other, which means $f(x)$ is not irreducible. Is this correct? Or am I missing pieces, or completely misunderstanding things? Thank you for your help.