GATE | Gate IT 2008 | Question 25

Consider the field C of complex numbers with addition and multiplication. Which of the following form(s) a subfield of C with addition and multiplication?
(S1) the set of real numbers
(S2) {(a + ib) | a and b are rational numbers}
(S3) {a + ib | (a2 + b2) ≤ 1}
(S4) {ia | a is real}
(A) only S1
(B) S1 and S3
(C) S2 and S3
(D) S1 and S2


Answer: (D)

Explanation: S3: is not closed
=> (0.3+0.4i)+(0.7+0.6i)=1+i
Both (0.3+0.4i) and (0.7+0.6i)
Complex numbers follows S3 but Addition of them does not follows
=>1+i, (a2+b2)<=1

=> 1+1 is not less than equal to 1
S4: {ia |a is real}
There is no multiplicative identity exists (that is 1)

S1: It is closed under both * and +
As real + real =real and real * real = real

S2: It is closed as rational + rational = rational and rational * rational = rational


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