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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.
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 ...
By making a modular multiplicative inverse table for the finite field and doing a lookup. By mapping to a composite field where inversion is simpler, and mapping back. By constructing a special integer (in case of a finite field of a prime order) or a special polynomial (in case of a finite field of a non-prime order) and dividing it by a. [7]
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]
One important application of Berlekamp's algorithm is in computing discrete logarithms over finite fields , where is prime and . Computing discrete logarithms is an important problem in public key cryptography and error-control coding .
The practical application of FEM is known as finite element analysis (FEA). FEA, as applied in engineering , is a computational tool for performing engineering analysis . It includes the use of mesh generation techniques for dividing a complex problem into smaller elements, as well as the use of software coded with a FEM algorithm.
In mathematics, local class field theory, introduced by Helmut Hasse, [1] is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of ...
The field F is uniquely determined by these properties, up to a field automorphism (i.e. essentially up to the notation of its elements). F is countable and contains a single copy of each of the finite fields GF(2 n); the copy of GF(2 n) is contained in the copy of GF(2 m) if and only if n divides m.