Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then $C=\{\alpha_1^p,\dots,\alpha_n^p\}$ is a basis.
Well, my first idea is to use the following map:
$T:\mathbb{F}\longrightarrow\mathbb{F}$ where $T(\alpha_i)=\alpha_i^p$ with $i=1,\dots,n$
Proving that $T$ is isomorphism between vector spaces the problem is solved, since that $T$ carries basis to basis.
$T(k\alpha_i)=kT(\alpha_i)$ for $k\in\mathbb{K}$.
Now i have to prove $T(\alpha_i+\alpha_j)=\alpha_i^p + \alpha_j^p$, surjectivity and injectivity of $T$, but at first, it does not look true.
Maybe I'm in the wrong way, because I'm not using the fact that $\mathbb{F}/\mathbb{K}$ is separable.