Which of the following pairs is an extension of fields?

Question

I keep getting this quiz question wrong but I don't know why the one I selected is wrong. Is the only answer that $Q(i)$ is the extension of R?

Answer 1

For apair $A,B$ to be an extension of fields, both $A$ and $B$ must be fields and either $A\subseteq B$ or $B\subseteq A$.

• $\Bbb Q(i)\not\subseteq \Bbb R$ because $i\notin \Bbb R$. $\Bbb R\not\subseteq \Bbb Q(i)$ because $\sqrt 2\notin\Bbb Q(i)$

• $\Bbb Z$ is not a field

• $\Bbb Q(i)\subset \Bbb C$

• $\Bbb Q\subset \Bbb R$

Hence options 3 and 4 are correct.