Field extensions: compute the degree of an extension.

Question

I'm stuck with this problem.

Let $F\subseteq E$ and $\gamma\in E$ is trascendental over $F$. Let $m$ be a positive integer.

Show that $[F(\gamma):F(\gamma^{m})]=m$, where $[\quad:\quad]$ is the degree of the extension.

So far, I've tried this: We know that $p(x)=x^{m}-\gamma^{m}\in F(\gamma^{m})[x]$, and $p(\gamma)=0$, so $[F(\gamma):F(\gamma^{m})]\leq m$. I tried to exhibit a set of l.i. vectors of $F(\gamma)$ over $F(\gamma^{m})$. But I failed. Any idea?

Answer 1

$1,\gamma, \ldots , \gamma^{m-1}$ are independent over $F(\gamma^{m})$