Discussion
Let F be a field with 128 elements, then which of the following is true ?
Answer: B
The true statement is F has no proper non-prime subfield.
\(128 = 2^{7}\), so F is \(\mathbf{F}_{2^{7}}\). The subfields of \(\mathbf{F}_{p^{n}}\) are exactly the fields \(\mathbf{F}_{p^{d}}\) with d dividing n. Since 7 is prime, the only divisors are 1 and 7: the subfields are the prime field \(\mathbf{F}_{2}\) and F itself. So F has no proper subfield other than the prime field.
A subfield with 8 elements would need \(3 \mid 7\), and one with 32 elements would need \(5 \mid 7\); neither holds. And a field with 128 elements does exist, since a finite field of order \(p^{n}\) exists for every prime p and every \(n \ge 1\).
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development