Solution
= Solution
Start with $S_0=\mathbb Q$. Given countable $S_n$, let $S_{n+1}$ be the field generated by $S_n\cup\{\sqrt{|x|}:x\in S_n\}$. It is countable because its elements are values of rational expressions in finitely many members of a countable set. Then $S=\bigcup_nS_n$ is countable and has (i)–(iii). Any other set with those properties contains every $S_n$ by induction, proving minimality and uniqueness.