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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 different branches of mathematics, primitive polynomial may refer to: Primitive polynomial (field theory), a minimal polynomial of an extension of finite fields;
A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of ...
The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p) = p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial. This ...
A polynomial P with coefficients in a UFD R is then said to be primitive if the only elements of R that divide all coefficients of P at once are the invertible elements of R; i.e., the gcd of the coefficients is one. Primitivity statement: If R is a UFD, then the set of primitive polynomials in R[X] is closed under
As before, let be a primitive th root of unity in (), and let () be the minimal polynomial over () of for all . The generator polynomial of the BCH code is defined as the least common multiple g ( x ) = l c m ( m c ( x ) , … , m c + d − 2 ( x ) ) . {\displaystyle g(x)={\rm {lcm}}(m_{c}(x),\ldots ,m_{c+d-2}(x)).}
The Conway polynomial C p,n is defined as the lexicographically minimal monic primitive polynomial of degree n over F p that is compatible with C p,m for all m dividing n.This is an inductive definition on n: the base case is C p,1 (x) = x − α where α is the lexicographically minimal primitive element of F p.
An important relation linking cyclotomic polynomials and primitive roots of unity is ∏ d ∣ n Φ d ( x ) = x n − 1 , {\displaystyle \prod _{d\mid n}\Phi _{d}(x)=x^{n}-1,} showing that x {\displaystyle x} is a root of x n − 1 {\displaystyle x^{n}-1} if and only if it is a d th primitive root of unity for some d that divides n .