I was reading through some field theory, and was wondering whether the minimal polynomial of a general element in a field extension L/K has degree less than or equal to the degree of the field extension? i.e.
$deg$(m$_{a}$) $\leq$ $[L:K]$ $\forall$ a $\in$ $L$
If it is true, then could anyone give me an idea as to how to prove this please?
Best Answer
$d=[L:K]$ is the dimension of $L$ when viewed as a vector space over $K$.
Consider $1,a,a^2,\dots,a^{d}$. This is a list of $d+1$ elements of $L$. Can they be linearly independent over $K$? What does it mean if they aren't?