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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

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    why downvoted??2012-12-19
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    I upvoted to counter the downvote which I also don't understand.2012-12-19
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    @BrunoStonek yes, thank you2012-12-19

2 Answers 2