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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.
GF(2) is the only field with this property (Proof: if x 2 = x, then either x = 0 or x ≠ 0. In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1. All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property).
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.
On the left, the lattice diagram of the field obtained from Q by adjoining the positive square roots of 2 and 3, together with its subfields; on the right, the corresponding lattice diagram of their Galois groups. In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory.
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] More narrowly, a Galois geometry may be defined as a projective space over a finite field. [2] Objects of study include affine and ...
The table of primitive polynomials shows how LFSRs can be arranged in Fibonacci or Galois form to give maximal periods. One can obtain any other period by adding to an LFSR that has a longer period some logic that shortens the sequence by skipping some states.
This field is called a finite field or Galois field with four elements, and is denoted F 4 or GF(4). [8] The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F 2 or GF(2) .
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory , so named in honor of ...