Let $F=\mathbb{Q}(2^{1/n})$. Prove that $[F:\mathbb{Q}]=n$.
I know that this is asking to prove the dimension of the vector space $F$ over the field of scalars $\mathbb{Q}$ is $n$, and I'm aware the basis should be $\{1,2^{1/n},2^{2/n},\dots,2^{(n-1)/n}\}$, but I'm having a hard time showing explicitly why these are linearly independent over $\mathbb{Q}$ and why they span.