Finite field

In abstract algebra, a finite field or Galois field is a field which contains only finitely many elements. Finite fields are important in cryptography and [coding theory]?. The finite fields are completely known, as will be described below.

Since every field of characteristic 0 contains the rationals and is therefore infinite, all finite fields have prime characteristic.

If p is a prime, the integers modulo p form a field with p elements, denoted by Z_p or F_p. Ever other field with p elements is isomorphic to this one.

If q = pⁿ is a prime power, then there exists up to isomorphism exactly one field with q elements, written as F_q. It can be constructed as follows: find an [irreducible polynomial]? f(T) of degree n with coefficients in F_p, then define F_q = F_p[T] / (f(T)). Here, F_p[T] denotes the ring of all polynomials with coefficients in F_p, and the quotient is meant in the sense of factor rings. The polynomial f(T) can be found by factoring the polynomial T ^q-T over F_p. The field F_q contains F_p as a subfield.

There are no other finite fields.

Example

The polynomial f(T) = T ² + T + 1 is irreducibe over F₂, and F₄ can therefore be written as the set {0, 1, t, t+1} where the multiplication is defined by t² + t + 1 = 0.