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 primaryirreducible (i.e.[[monic powerpolynomial]]s of andegree irreducible)exactly n is the number of [[monicNecklace polynomial(combinatorics)#Aperiodic necklaces|aperioidic necklaces]], given by [[Necklace polynomial|Moreau's ofnecklace-counting degreefunction]] M<sub>''q''</sub>(''n''). overThe closely related necklace function '''F'N''<sub>''q''</sub>(''n'') iscounts equalmonic topolynomials thewhich [[Necklaceare primary (combinatoricsa power of an irreducible)|necklacecountingof polynomial]]degree ''Nn'' 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> Alternatively, one may consider this as the number of irreducible polynomials of all degrees d which divide n.
The number of irreducible monic polynomials of degree exactly n is the number of [[Necklace (combinatorics)#Aperiodic necklaces|aperioidic necklaces]], given by [[Necklace polynomial|Moreau's polynomial]] M<sub>''q''</sub>(''n'').
