Characteristic of a field extension

Question

Let $\mathbb{F}_p$ be a field. Is it correct that the field extensions are $\mathbb{F}_{p^n}$, with $n \in \mathbb{N}$? And does the characteristic of every $\mathbb{F}_{p^n}$ equal $p$?

Answer 1

Indeed, all the finite field extensions are of the form $\mathbb F_{p^n}$ (up to isomorphism), and all these fields have characteristic $p$. Note that in general, the characteristic of a field is always equal to that of its prime subfield. Another way of looking at this: the only way you can have a morphism between two fields (remember that they are always injective) is if the two fields have the same characteristic; embedding is definitely a morphism.