Irreducible polynomials over finite fields are also useful for [[Pseudorandom]] number generators using feedback shift registers and [[discrete logarithm]] over '''F'''<sub>2<sup>''n''</sup></sub>.
The number of irreducible [[monic polynomial]]s of degree exactly n over '''F'''<sub>''q''</sub> is the number of [[Necklace (combinatorics)#Aperiodic necklaces|aperioidic necklaces]], given by [[Necklace polynomial|Moreau's necklace-counting function]] ''M''<sub>''q''</sub>(''n''). The closely related necklace function ''N''<sub>''q''</sub>(''n'') counts monic polynomials of degree ''n'' which are primary (a power of an irreducible) of degree ''n'' over '''F'''<sub>''q''</sub>; or alternatively the irreducible polynomials of all degrees d which divide n.<ref>Christophe Reutenauer, ''Mots circulaires et polynomes irreductibles'', Ann. Sci. math Quebec, vol 12, no 2, pp. 275-285</ref>
