Minimal polynom of an element

Question

If for example i take an extension $E/K$ over a field $K$ and an element $\alpha \in E - K $ and I find a polynomial $P$ such that $P(\alpha)=0$. How can I show this polynomial is the minimal polynomial? Is it enough if I show this polynomial is irreducible over $K$ or if I show that $deg(P)=[K(\alpha ): K]$ ? Thanks for your answers

Answer 1

Yes: if $\alpha$ is a root of $P(x)\in K[x]$, then $P(x)$ is a minimal poynomial of $\alpha$ if and only if $P(x)$ is irreducible in $K[x]$. And then $\deg P(x)=\bigl[K(\alpha):K\bigr]$.