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 aperiodic necklaces 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).

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