Prime subfield of a field is $mathbb{Z}/pmathbb{Z}$ or $mathbb{Q}$.

Question

I can't see the phrase "$F$ contains a subfield isomorphic either $mathbb{Q}$ (the field of fractions of $mathbb{Z})$ or to $mathbb{F}_{p}$"
I have the following: By 1 Isomorphism's theorem, $mathbb{Z}/ker(varphi)simeq Im(varphi)subset F$. Now, if $ch(F)=0$ then $mathbb{Z}/left{0right}simeq Im(varphi)$ but i don't see that $mathbb{Q}$ is contained in $F$ (I only see that a copy of $mathbb{Z}$ us contained in $F$)
In the examples, why the prime subfield of $mathbb{Q},mathbb{R}$ is $mathbb{Q}$?

J. W. Tanner · Answer

You acknowledged that, if the characteristic of a field $F$ is $0$, then a copy of $mathbb Z$, say $varphi(mathbb Z)$, is in $F$.
But any $qin mathbb Q$ can be expressed as $nm^{-1}$ with $n,min mathbb Z$,
and, because $F$ is a field, $varphi(n),varphi(m)in  F$ means $varphi(m)^{-1}in F$ and $varphi(n)varphi(m)^{-1}in F.$

Prime subfield of a field is $mathbb{Z}/pmathbb{Z}$ or $mathbb{Q}$.

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