Irreducible Polynomial

A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field.

For example, in the field of rational polynomials  (i.e., polynomials  with rational coefficients),  is said to be irreducible if there do not exist two nonconstant polynomials  and  in  with rational coefficients such that

(Nagell 1951, p. 160). Similarly, in the finite field GF(2),  is irreducible, but  is not, since (mod 2).

Irreducible polynomial checking is implemented in the Wolfram Language as IrreduciblePolynomialQ[poly].

In general, the number of irreducible polynomials of degree  over the finite field GF() is given by

where  is the Möbius function.

The number of irreducible polynomials of degree  over GF(2) is equal to the number of -bead fixed aperiodicnecklaces of two colors and the number of binary Lyndon words of length . The first few numbers of irreducible polynomial (mod 2) for , 2, ... are 2, 1, 2, 3, 6, 9, 18, ... (OEIS A001037). The following table lists the irreducible polynomials (mod 2) of degrees 1 through 5.

	irreducible polynomials
1	,
2
3	,
4	, ,
5	, , , , ,

The possible polynomial orders of th degree irreducible polynomials over the finite field GF(2) listed in ascending order are given by 1; 3; 7; 5, 15; 31; 9, 21, 63; 127; 17, 51, 85, 255; 73, 511; ... (OEIS A059912).

REFERENCES:

Marsh, R. Tables of Irreducible Polynomials of GF(2) through Degree 19. Washington, DC: U. S. Dept. Commerce, 1957.

Nagell, T. "Irreducibility of the Cyclotomic Polynomial." §47 in Introduction to Number Theory. New York: Wiley, pp. 160-164, 1951.

Ruskey, F. "Information on Primitive and Irreducible Polynomials." http://www.theory.csc.uvic.ca/~cos/inf/neck/PolyInfo.html.

Sloane, N. J. A. Sequences A001037/M0116 and A059912 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Figure M0564 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.