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Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
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.
If is an irreducible polynomial of prime degree with rational coefficients and exactly two non-real roots, then the Galois group of is the full symmetric group. [2] For example, f ( x ) = x 5 − 4 x + 2 ∈ Q [ x ] {\displaystyle f(x)=x^{5}-4x+2\in \mathbb {Q} [x]} is irreducible from Eisenstein's criterion.
In this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
Given a prime number q and prime power q m with positive integers m and d such that d ≤ q m − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = q m − 1 and distance at least d is constructed by the following method. Let α be a primitive element of GF(q m).
If g is a primitive root modulo p k, then g is also a primitive root modulo all smaller powers of p. If g is a primitive root modulo p k, then either g or g + p k (whichever one is odd) is a primitive root modulo 2 p k. [14] Finding primitive roots modulo p is also equivalent to finding the roots of the (p − 1)st cyclotomic polynomial modulo p.
To find a primitive polynomial of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find primitive polynomials of very high order. Such primitive trinomials are used in pseudorandom number generators with very large periods such as the Mersenne twister , generalized shift register and ...
Primitive polynomial (field theory), a minimal polynomial of an extension of finite fields Primitive polynomial (ring theory) , a polynomial with coprime coefficients Topics referred to by the same term