Prime subfield is either isomorphic to $\mathbb{Q}$ or $F_p$ I'm trying to prove the following statement:
Let $F$ be a field. The intersection of all subfields of $F$ is a subfield which is isomorphic to $\mathbb{Q}$ if $\operatorname{char}(F)=0$, and isomorphic to $F_p$ if $\operatorname{char}(F)=p$.
I assume I need to set up a injective ring homomorphism $\varphi\colon \mathbb{Q}\to F.$ I don't really know where to go apart from that though. 

Define a map $f:\Bbb{Z}\to{F}$  by $f(n)=n.e$ , where $e$ is the unit element of $F$.
$f$ is homomorphism:
$$f(m+n)=(m+n).e=m.e+n.e=f(m)+f(n)$$
and
$$f(mn)=(mn).e=(m.e)(n.e)=f(m)f(n)$$.
By Fundamental theorem of homomorphism,
$$f(\Bbb{Z})\cong \frac{\Bbb{Z}}{\operatorname{Ker}{f}}\subseteq F$$.
Where $\operatorname{Ker}{f}$ is the ideal of $\Bbb{Z}$, in fact principal ideal of $\Bbb{Z}$.Then $\operatorname{Ker}f=<q> $, for some $q\in \Bbb{Z}$.
Now consider the needed cases and you are done!
