My question is
Prove that: Any field containing $Z[i]$ as a subring contains $Q[i]$ as a subfield.
I don't know how to prove this question. Please if someone could prove it for me.
It seems when I asked this question before it was not clear enough so a direct approach. If you'll still find my question inappropriate then please tell me how to frame one. I would really like the added help. Thank you.
Let $F$ be a field which contains the integers $a$ and $b$ with $b\ne 0$. Then by definition of a field $F$ also contains $1/b$, hence contains $a/b$. So any field which contains $\mathbb Z$ also contains $\mathbb Q$.
It follows that if $F$ contains $\mathbb Z$ and $i$ then $F$ contains $\mathbb Q$ and $i$, hence contains $\mathbb Q(i)$.