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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 Primitive polynomial (ring theory) , a polynomial with coprime coefficients
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 ...
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
A monic irreducible polynomial of degree n having coefficients in the finite field GF(q), where q = p t for some prime p and positive integer t, is called a primitive polynomial if all of its roots are primitive elements of GF(q n). [2] [3] In the polynomial representation of the finite field, this implies that x is a primitive element.
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.
If K / F is a Galois extension and x in K generates a normal basis over F, then x is free in K / F. If x has the property that for every subgroup H of the Galois group G, with fixed field K H, x is free for K / K H, then x is said to be completely free in K / F. Every Galois extension has a completely free element. [2]