# Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$

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.

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Please see here for a guide to writing math with MathJax, and see here for a guide to formatting posts with Markdown. – Zev Chonoles Aug 11 '13 at 5:25

Hint:

Consider the subfield generated by the identity, that is, the smallest subfield that contains the identity.

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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.

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