Let $F$ be a finite field where $char(F) = p$. I encountered the following preposition in my textbook (given the stated conditions): "We have an injective homomorphism $\varphi: \mathbb{F}_p \longrightarrow F$." My question: why is this $\varphi$ not an isomorphism? Since all finite fields of order $n$ (where $n$ is prime) are isomorphic to $\mathbb{F}_n$, I don't see how our $\varphi$ would behave any differently (I suspect I have a misunderstanding of $char(F)$, but I'm not sure what it is).
Any and all help is greatly appreciated.
Best Answer
Note that if $char(F) = p$, this does not imply that $|F| = p$. Rather, it implies that $|F|=p^k$ for some $k$.