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Which of the following numbers can be the order (i.e., the number of elements) of a finite field?

A. 125
B. 69
C. 93
D. 225
E. 199
F. 811
G. 116
H. 283
I. 256
J. 106

Answer :

Final answer:

The order of a finite field must be a power of a prime number. From the given list, the numbers 125 (which is 5^3) and 256 (which is 2^8) fit this criterion and can be the order of a finite field.

Explanation:

The question is about the possible orders of a finite field. In mathematics, specifically in the context of finite fields or Galois fields, the order of a finite field must be a power of a prime number. This means that the number of elements in the field, its order, must be an expression of the form p^n where p is a prime number and n is a positive integer.

To determine which of the given numbers can be the order of a finite field, we identify which of them are powers of a prime number:

  • 125 is 5^3, so it can be the order of a finite field.
  • 256 is 2^8, so it can also be the order of a finite field.
  • The other numbers listed (69, 93, 225, 199, 811, 116, 283, 106) are not powers of a prime number, so they cannot be the order of a finite field.

Therefore, the numbers that can be the order of a finite field from the given list are 125 and 256.

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