Consider the field C of complex numbers with addition and multiplication.…

Answer: S1 and S2 are subfields; S3 and S4 are not.

• S1 — the real numbers: The real numbers are closed under addition and multiplication, contain 0 and the multiplicative identity 1, and every nonzero real has a real multiplicative inverse. Therefore the real numbers form a subfield of the complex numbers.

• S2 — complex numbers with rational components: If a and b are rational, sums and products of such numbers have rational components because the rationals are closed under addition and multiplication. The multiplicative identity 1 = 1+0i is in the set. For a+ib ≠ 0 with a,b rational, the inverse is (a-ib)/(a^2+b^2); since a^2+b^2 is a nonzero rational, dividing rational numerators by a nonzero rational gives rational components. Hence this set is a field (often written Q(i)) and so is a subfield of the complex numbers.

• S3 — numbers with a^2 + b^2 ≤ 1: This set is not a field. It fails to contain multiplicative inverses in general. Example: 1/2 (which has modulus 1/2 ≤ 1) lies in the set, but its multiplicative inverse 2 has modulus 2 > 1 and so is not in the set. The set also fails closure under addition in some cases.

• S4 — purely imaginary numbers {i a | a real}: This set does not contain the multiplicative identity 1, so it cannot be a subfield. Also products of two nonzero purely imaginary numbers are real (for example (i)(i) = -1), but most real numbers are not in the purely imaginary set, so it fails closure under multiplication as a field would require.

Conclusion: The subfields among the listed sets are the real numbers and the set of complex numbers with rational real and imaginary parts.