Proof: Degree of a field extension

Question

Let $n\in\mathbb{N}$ and $a\in\mathbb{Q}$, so that the polynomial $f:=t^n-a\in\mathbb{Q}[t]$ is irreducible in $\mathbb{Q}[t]$. Let $\alpha$ be a root of $f$ and $m\in\mathbb{N}$ a divisor of $n$. I have to show that [$\mathbb{Q}(\alpha^m):\mathbb{Q}]=n/m$. I know that this means that [$\mathbb{Q}(\alpha):\mathbb{Q}]=n$ but i don´t know how to move on from here. Any help is very much appreciated.

Answer 1

Hint 1 $$n=[\mathbb{Q}(\alpha):\mathbb{Q}]=[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^m)]\cdot[\mathbb{Q}(\alpha^m):\mathbb{Q}]$$

Hint 2: $\alpha$ is a root of $X^m-\alpha^m$ which is a polynomial with coefficients in $\mathbb{Q}(\alpha^m)$. Therefore $$[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^m)] \leq m$$

Hint 3: $\alpha^m$ is a root of $X^\frac{m}{n}-a \in \mathbb Q[X]$.