minimum polynomial definition

Question

I was wondering if anyone could define the term "$m(x)$ is a minimum polynomial of $\alpha$ over $K$".

I thought it means $m(x)$ is a monic polynomial in $K$ such that $m(\alpha)=0$?

Answer 1

Clearly, if $p(x)$ is a polynomial such that $p(\alpha)=0$, then any polynomial $f(x)$ with $p|f$ has $f(\alpha)$ too. So, there are lots of polynomials that kills $\alpha$. However, if we ask the polynomial to be monic and of the smallest possible order (such that it still kills $\alpha$!), then there is only one (exercise), which is called the minimal polynomial. In this point of view, you can feel that a minimal polynomial is a very canonical object that relates to your element $\alpha$.

When $\alpha$ is in the base field, $x-\alpha$ is the minimal polynomial of $\alpha$ since it kills $\alpha$ directly. However, when $\alpha$ is not in the base field, any polynomials of order 1 cannot kill $\alpha$ (easy exercise). Thus the degree of the minimal polynomial somehow "measures" how far $\alpha$ is from the base field. The further $\alpha$ stays away from the base field, the higher the order should be. We thus define the degree of $\alpha$ to be the order of the minimal polynomial.

Any algebraic number stays "finitely far away" from $\mathbb{Q}$, while the others (transcendentals like $e$ or $\pi$) stays "infinitely far away"!