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In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).
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.
In fact, if P is an irreducible factor over GF(p) of X q − X, its degree divides n, as its splitting field is contained in GF(p n). Conversely, if P is an irreducible monic polynomial over GF( p ) of degree d dividing n , it defines a field extension of degree d , which is contained in GF( p n ) , and all roots of P belong to GF( p n ) , and ...
Mysterious Girlfriend X (Japanese: 謎の彼女X, Hepburn: Nazo no Kanojo Ekkusu) is a Japanese manga written and illustrated by Riichi Ueshiba.It was originally published as a one-shot story in 2004 before becoming a serialized comic in Kodansha's seinen manga magazine Monthly Afternoon from March 2006 to September 2014, with its chapters collected in 12 tankōbon volumes.
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 ...
Elements of GF(p n) may be represented as polynomials of degree strictly less than n over GF(p). Operations are then performed modulo m(x) where m(x) is an irreducible polynomial of degree n over GF(p), for instance using polynomial long division. Addition is the usual addition of polynomials, but the coefficients are reduced modulo p.
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If f(x) is a permutation polynomial defined over the finite field GF(q), then so is g(x) = a f(x + b) + c for all a ≠ 0, b and c in GF(q). The permutation polynomial g(x) is in normalized form if a, b and c are chosen so that g(x) is monic, g(0) = 0 and (provided the characteristic p does not divide the degree n of the polynomial) the ...