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The finite field with p n elements is denoted GF(p n) and is also called the Galois field of order p n, in honor of the founder of finite field theory, Évariste Galois. GF(p), where p is a prime number, is simply the ring of integers modulo p. That is, one can perform operations (addition, subtraction, multiplication) using the usual operation ...
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z. Notations Z 2 and Z 2 {\displaystyle \mathbb {Z} _{2}} may be encountered although they can be confused with the notation of 2 -adic integers .
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]
The finite field (or Galois field) GF(125) = GF(5 3) has degree 3 over its subfield GF(5). More generally, if p is a prime and n, m are positive integers with n dividing m, then [GF(p m):GF(p n)] = m/n. The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental ...
The order of a finite field is always a prime or a power of prime. For each prime power q = p r, there exists exactly one finite field with q elements, up to isomorphism. This field is denoted GF(q) or F q. If p is prime, GF(p) is the prime field of order p; it is the field of residue classes modulo p, and its p elements are denoted 0, 1 ...
Let L = GF(q n) be a finite extension of a finite field K = GF(q).Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e. [4]
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q − 1) th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as α i for some natural number i.