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e. 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 monic irreducible polynomial x 8 + x 4 + x 3 + x + 1 over GF(2) is not primitive. Let λ be a root of this polynomial (in the polynomial representation this would be x), that is, λ 8 + λ 4 + λ 3 + λ + 1 = 0. Now λ 51 = 1, so λ is not a primitive element of GF(2 8) and generates a multiplicative subgroup of order 51. [5]
The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF (2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a ...
Let α be a primitive element of GF(q m). For any positive integer i, let m i (x) be the minimal polynomial with coefficients in GF(q) of α i. The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x ...
Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a ...
A finite field or Galois field is a field with a finite order (number of elements). The order of a finite field is always a prime or a power of prime. For each prime power q = pr, there exists exactly one finite field with q elements, up to isomorphism. This field is denoted GF (q) or Fq.
Calculus. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [1][2][3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. It is provable in many ways by using other derivative rules.
Over GF(3) the polynomial x 2 + 1 is irreducible but not primitive because it divides x 4 − 1: its roots generate a cyclic group of order 4, while the multiplicative group of GF(3 2) is a cyclic group of order 8. The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α.