In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field . This means that a polynomial of degree with coefficients in is a primitive polynomial if it is monic and has a root in such that <math>\{0,1,\alpha, \alpha^2,\alpha^3, \ldots \alpha^{p^m-2}\}</math> is the entire field . This implies that is a primitive ()-root of unity in .
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 GF(2), 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 (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 is .
- A primitive polynomial of degree has different roots in , which all have order , meaning that any of them generates the multiplicative group of the field.
- Over GF(p) there are exactly primitive elements and primitive polynomials, each of degree , where is Euler's totient function.
- The algebraic conjugates of a primitive element in are , , , …, and so the primitive polynomial has explicit form . That the coefficients of a polynomial of this form, for any in , not necessarily primitive, lie in 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 .
Examples
Over the polynomial is irreducible but not primitive because it divides : its roots generate a cyclic group of order 4, while the multiplicative group of is a cyclic group of order 8. The polynomial , on the other hand, is primitive. Denote one of its roots by . Then, because the natural numbers less than and relatively prime to are 1, 3, 5, and 7, the four primitive roots in are , , , and . The primitive roots and are algebraically conjugate. Indeed . The remaining primitive roots and are also algebraically conjugate and produce the second primitive polynomial: .
For degree 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 . Denoting one of its roots by , the algebraically conjugate elements are and . The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements with relatively prime to 26:
:<math>\begin{align}x^3+2x+1 & = (x-\gamma)(x-\gamma^3)(x-\gamma^9)\\
x^3+2x^2+x+1 &= (x-\gamma^5)(x-\gamma^{5\cdot3})(x-\gamma^{5\cdot9}) = (x-\gamma^5)(x-\gamma^{15})(x-\gamma^{19})\\
x^3+x^2+2x+1 &= (x-\gamma^7)(x-\gamma^{7\cdot3})(x-\gamma^{7\cdot9}) = (x-\gamma^7)(x-\gamma^{21})(x-\gamma^{11})\\
x^3+2x^2+1 &= (x-\gamma^{17})(x-\gamma^{17\cdot3})(x-\gamma^{17\cdot9}) = (x-\gamma^{17})(x-\gamma^{25})(x-\gamma^{23}).
\end{align}</math>
Applications
Field element representation
Primitive polynomials can be used to represent the elements of a finite field. If α in GF(p<sup>m</sup>) is a root of a primitive polynomial F(x), then the nonzero elements of GF(p<sup>m</sup>) are represented as successive powers of α:
:<math>
\mathrm{GF}(p^m) = \{ 0, 1= \alpha^0, \alpha, \alpha^2, \ldots, \alpha^{p^m-2} \} .
</math>
This allows an economical representation in a computer of the nonzero elements of the finite field, by representing an element by the corresponding exponent of <math>\alpha.</math> This representation makes multiplication easy, as it corresponds to addition of exponents modulo <math>p^m-1.</math>
Pseudo-random bit generation
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 , 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 pseudo-random bits before repeating the same sequence.
CRC codes
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 for a degree n primitive polynomial.
Primitive trinomials
A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms: . 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 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 . 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 . This can be used to create a pseudo-random number generator of the huge period ≈ .