Editing Finite field (section)

=== Number of monic irreducible polynomials of a given degree over a finite field ===
The number {{math|''N''(''q'', ''n'')}} of monic irreducible polynomials of degree {{math|''n''}} over 
{{math|GF(''q'')}} is given by<ref>{{harvnb|Jacobson|2009|loc=§4.13}}</ref>
<math display="block">N(q,n)=\frac{1}{n}\sum_{d\mid n} \mu(d)q^{n/d},</math>
where {{math|''μ''}} is the [[Möbius function]]. This formula is an immediate consequence of the property of {{math|''X''<sup>''q''</sup> − ''X''}} above and the [[Möbius inversion formula]].

By the above formula, the number of irreducible (not necessarily monic) polynomials of degree {{math|''n''}} over {{math|GF(''q'')}} is {{math|(''q'' − 1)''N''(''q'', ''n'')}}.

The exact formula implies the inequality
<math display="block">N(q,n)\geq\frac{1}{n} \left(q^n-\sum_{\ell\mid n, \ \ell \text{ prime}} q^{n/\ell}\right);</math>
this is sharp if and only if {{math|''n''}} is a power of some prime.
For every {{math|''q''}} and every {{math|''n''}}, the right hand side is positive, so there is at least one irreducible polynomial of degree {{math|''n''}} over {{math|GF(''q'')}}.