If $F$ is a field with characteristic $p$ and $\alpha$ is element of an extension of $F$, $F(\alpha)$ is also a field with characteristic $p$?
At the first glance, it seems obvious, because $F(\alpha)$ is the field generated by $F$ and $\alpha$. The problem is this element $\alpha$ which doesn't need to be in an extension of characteristic $p$.
I'm a little confused, I need help.
Thanks