Finite field

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A finite field is a field with only finitely many elements.

Some key facts about finite fields include:

  • The characteristic of a finite field must be a prime number.
  • The size of a finite field must be a prime power. In fact, it must be a power of the prime number that is the characteristic of the field.
  • For any prime power, there is a unique (up to isomorphism) finite field whose size equals that prime power.

Combining the above key facts, we denote, for any prime power q, the unique finite field of size q by the symbols \mathbb{F}_q or GF(q). (Note that GF stands for Galois field in recognition of Galois's pioneering work in field theory).

Particular cases

Field size q Underlying prime p (field characteristic) \log_pq, i.e, the number r such that q = p^r Field
2 2 1 field:F2
3 3 1 field:F3
4 2 2 field:F4
5 5 1 field:F5
7 7 1 field:F7
8 2 3 field:F8
9 3 2 field:F9
11 11 1 field:F11
13 13 1 field:F13
16 2 4 field:F16
17 17 1 field:F17