My book is the GTM$173$, Field and Galois Theory written by Patrick Morandi.
F$_p$ represents $Z/pZ$ ($Z$ is the integer set) in my textbook. And $F$ is a field with characteristic $p$. The author writes that 'we can view F$_p$ as subfield of $F$'. But he doesn't explain clearly what's the embedding is. I think one way is to map $1+pZ$ in $Z/pZ$ to $e$, the identity in $F$.
Also, in the picture I upload, he defines a mapping and claim $a^p=a$ iff $a\in$ F$_p$. I don't know why. Can we prove that if $a^p=a$ then $a$ must be one of the element from {$0,e,2e,3e...(p-1)e$}???
Yes, the prime field of any field is the subfield generated by the multiplicative identity, $1$ of the field.
Note that it can only be $\Bbb Q$ or $\Bbb F_p=\Bbb Z/p\Bbb Z$.
The order of any group element divides the order of the group. Apply this to the multiplicative group of $\Bbb F_p$ to see $a^{p-1}=1$ for all nonzero $a\in\Bbb F_p$, and thus $a^p=a$ follows.
Conversely, the polynomial $X^p-X$ can have at most $p$ roots, since we work with a field, but each element of $\Bbb F_p$ is already a root, so can't have more.