Шектелi емес даланың примитивті элементінің минималды полиномы

Minimal polynomial of a primitive element in a finite field
polynomials such that the greatest common divisor of the coefficients is 1

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 1=GF(p) = 'Z'/p'Z' 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)). This implies that α is a primitive (p^(m) − 1) root of unity in GF(p^(m)).

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 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 such that F(x) divides xn − 1 is 1=n = pm − 1. A primitive polynomial of degree m has m different roots in GF(p^(m)), which all have order p^(m) − 1, meaning that any of them generates the multiplicative group of the field. Over GF(p) there are exactly φ(p^(m) − 1) primitive elements and φ(p^(m) − 1) / m primitive polynomials, each of degree m, where φ is Euler's totient function. The algebraic conjugates of a primitive element α in GF(p^(m)) are α, , , , and so the primitive polynomial F(x) has explicit form That the coefficients of a polynomial of this form, for any α in GF(p^(n)), not necessarily primitive, lie in GF(p) follows from the property that the polynomial is invariant under application of the Frobenius automorphism to its coefficients (using ) and from the fact that the fixed field of the Frobenius automorphism is GF(p).

Over GF(3) the polynomial x^(2) + 1 is irreducible but not primitive because it divides x^(4) − 1: its roots generate a cyclic group of order 4, while the multiplicative group of GF(3^(2)) is a cyclic group of order 8. The polynomial x^(2) + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to are 1, 3, 5, and 7, the four primitive roots in GF(3^(2)) are α, , , and The primitive roots α and α^(3) are algebraically conjugate. Indeed The remaining primitive roots α^(5) and are also algebraically conjugate and produce the second primitive polynomial:
For degree 3, GF(3^(3)) has primitive elements. As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are primitive polynomials of degree 3. One primitive polynomial is x^(3) + 2x + 1. Denoting one of its roots by γ, the algebraically conjugate elements are γ^(3) and γ^(9). The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements γ^(r) with r relatively prime to 26:

Primitive polynomials over GF(2), the field with two elements, can be used for pseudorandom bit generation. In fact, every linear feedback shift register with maximum cycle length (which is 2n − 1, where n is the length of the linear feedback shift register) may be built from a primitive polynomial. In general, for a primitive polynomial of degree m over GF(2), this process will generate 2m − 1 pseudo random bits before repeating the same sequence.

The cyclic redundancy check (CRC) is an error detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see Mathematics of CRC. Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of 2n − 1 for a degree n primitive polynomial.

A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms: xr + xk + 1. Their simplicity makes for particularly small and fast linear feedback shift registers. A number of results give techniques for locating and testing primitiveness of trinomials. For polynomials over GF(2), where 2r − 1 is a Mersenne prime, a polynomial of degree r is primitive if and only if it is irreducible. (Given an irreducible polynomial, it is not primitive only if the period of x is a non trivial factor of 2r − 1. Primes have no non trivial factors.) Although the Mersenne Twister pseudo random number generator does not use a trinomial, it does take advantage of this. Richard Brent has been tabulating primitive trinomials of this form, such as x74207281 + x30684570 + 1. This can be used to create a pseudo random number generator of the huge period 274207281 − 1 ≈ 3.