If Iₘ denotes the set of integers modulo m, then the following are fields with respect to the…
The integers modulo m form a field exactly when m is prime, because only then does every non-zero element have a multiplicative inverse. 23, 29 and 31 are primes, so Z₂₃,…
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If Iₘ denotes the set of integers modulo m, then the following are fields with respect to the operations of addition modulo m and multiplication modulo m :
(i) Z₂₃
(ii) Z₂₉
(iii) Z₃₁
(iv) Z₃₃
Then
Answer and explanation
Correct answer C
The integers modulo m form a field exactly when m is prime, because only then does every non-zero element have a multiplicative inverse. 23, 29 and 31 are primes, so Z₂₃, Z₂₉ and Z₃₁ are fields. 33 = 3 × 11 is composite: 3 × 11 ≡ 0 (mod 33), so Z₃₃ has zero divisors and is only a ring.
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UGC NET Computer Science and Applications