Finite Field

In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and quantum error correction. The finite fields are classified by size; there is exactly one finite field up to isomorphism of size pk for each prime p and positive integer k. Each finite field of size q is the splitting field of the polynomial xq-x, and thus the fixed field of the Frobenius endomorphism which takes x to xq. Similarly, the multiplicative group of the field is a cyclic group. Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field. Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type. Finite fields are an active area of research, including recent results on the Kakeya conjecture and open problems on the size of the smallest primitive root.

Finite fields appear in the following chain of class inclusions:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.