galois theory - Extension field

Let $E$ an extension field of $k$ of grade $n$. I want to know if for $\alpha\in E$ the minimal polynomial of $\alpha$ has degree $r\leq n$. I think is true, but i could use some help

Answer

Conclude the desired result from the following:

Observation 1: Let $W$ be a subspace of a finite dimensional vector space $V$. Then $$\dim W \leq \dim V.$$