Finite field

related topics
{math, number, function}

In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. 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:


Full article ▸

related documents
Inverse function
Naive set theory
Word problem for groups
Cantor set
Inner product space
Limit superior and limit inferior
Peano axioms
Matrix multiplication
Zermelo–Fraenkel set theory
Computational complexity theory
Markov chain
Topological space
Collatz conjecture
Pythagorean theorem
Mathematical induction
Bra-ket notation
Hash table
Cauchy sequence
Recurrence relation
Forcing (mathematics)
Johnston diagram
Binary search tree
Non-standard analysis
Principal components analysis