Here is my current understanding of what a minimal polynomial is:
$p(x)$ is the minimal polynomial of $a$ over a field $\mathbb{F}$ if $p(a) = 0$ and $p(x)$ is the lowest degree irreducible, monic polynomial in $\mathbb{F}[x]$ for which this is true.
Is this exactly correct? Please tell me if not.
Now my question is, what are minimal polynomials used for? What applications does it have in algebra research? Does it tell us anything about the field we are working in?