Properties
Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible.
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over the field of two elements, 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.
An irreducible polynomial of degree m, F(x) over GF(p) for prime p, is a primitive polynomial if the smallest positive integer n such that F(x) divides xn - 1 is n = pm − 1.
Over GF(pm) there are exactly φ(pm − 1)/m primitive polynomials of degree m, where φ is Euler's totient function.
The roots of a primitive polynomial all have order pm − 1.
Read more about this topic: Primitive Polynomial (field Theory)
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