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Problem 46 (Every Field Contains A Subfield Isomorphic To $\mathbb{Z}_p$ or $\mathbb{Q}$)
Suppose that $F$ is a field.
- If $F$ has prime characteristic $p$, show that $F$ contains a subfield isomorphic to $Z_p$.
- If $F$ has characteristic 0, show that $F$ contains a subfield isomorphic to $\mathbb{Q}$.
Hint:You'll want to use the previous problem. If $p$ is prime, the previous problem should give this quickly. If the characteristic is zero, you'll need to use the copy of $Z$ sitting inside $F$ to create the subfield $Q$.
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