TITLE: Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$
QUESTION [5 upvotes]: Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$.

Could you help me prove this theorem?
My professor introduced this theorem during the lecture but didn't prove it.

REPLY [3 votes]: Let $F$ be a field. Define the map $f:\mathbb{Z}\to F$ by $f(0)=0_F$, $f(n)=f(n-1)+1_F$ if $n> 0$ and $f(m)=f(m+1)-1_F$ if $m<0$.
Note that $f$ is a ring homomorphism.
Also note that $\ker(f)=\operatorname{char}(F)\mathbb{Z}$. Use the fundamental theorem for ring isomorphisms.