Prove or disprove that $[L:K] = \deg(p)$

Question

Let $L/K$ be a finite extension and let $\beta \in L$. If $p$ is the minimal polynomial of $\beta$ then is it true that $[L:K] = \deg(p)$? If not, give a counterexample.

Can someone help me figure out if this is true or not?

Answer 1

Let $K=\mathbb{Q}(i,\sqrt{2})$ and consider the extension $K/\mathbb{Q}$. Consider the minimal polynomial of $i$ over $\mathbb{Q}$, which is $p(x)=x^{2}+1$, i.e. $\deg(p)=2$. But $$[K:\mathbb{Q}]=4\neq 2=\deg(p).$$