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). A polynomial can be tested to see if it is irreducible using the Mathematica function

  IrreducibleQ[p_,n_] := SameQ[Factor[p, Modulus->n], Expand[p]]

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, ... (Sloane's A001037). The following table lists the irreducible polynomials (mod 2) of degrees 1 through 5.

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