A field is an algebraic structure over two operations $+$ and $*$ if :
1. It is closed under both $+$ and $*.$
2. $+$ and $*$ are both commutative and associative. $(+$ and $*$ in this question are already commutative and associative, so no need to check$)$
3. Existence of additive identity$(0)$ and multiplicative identity $(1)$
4. Existence of additive and multiplicative inverses for each non-zero element.
5. Distributive property of $*$ over $+$ (This is also satisfied in question)
So, for each option, we have to check only properties $1, 3$ and $4.$
(S1) : set of all real numbers
1. Closed : Yes, $\text{real}+\text{real} = \text{real}, \text{real}*\text{real}=\text{real}$
3. Additive and multiplicative identity : Yes, $0$ and $1$ are real numbers
4. Additive and multiplicative inverse for each non-zero element: Yes, for any real number $a,$ additive inverse is $-a,$ which is also a real number, and multiplicative inverse is $1/a,$ which is also a real number.
(S2) : $\{(a + ib) \mid a$ and $b$ are rational numbers$\}$
1. Closed : Yes, $\text{rational}+\text{rational}=\text{rational}, \text{rational}*\text{rational}=\text{rational}$
3. Additive and multiplicative identity : Yes, $0+0i$ (additive identity) and $1+0i$ (multiplicative identity) belong to given set.
4. Additive and multiplicative inverse for each non-zero element : Additive inverse is $-a-ib,$ which belongs to given set. Multiplicative identity is $\frac{1}{a+ib} = \frac{a-ib}{a^2+b^2} = \frac{a}{a^2+b^2}+i\frac{-b}{a^2+b^2}$, which also belongs to given set.
(S3) : $\{a + ib \mid (a^2 + b^2) \leq 1\}$
1. Closed : No, for example : $(0.3+0.4i) + (0.7 + 0.6i) = 1 + i.$ Here, both complex numbers which were added were in the given set, but resultant complex number is not.
(S4) : $\{ia \mid a$ is real$\}$
Here, this set does not contain $1$ (multiplicative identity)
So, (S1) and (S2) are subfields of C. Option D) is correct.
