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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.
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
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.
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials.
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 ...
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.
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.