Editing Finite field (section)

=== Irreducible polynomials of a given degree ===
The polynomial
<math display="block">X^q-X</math> 
factors into linear factors over a field of order {{math|''q''}}. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order {{math|''q''}}.

This implies that, if {{math|1=''q'' = ''p<sup>n</sup>''}} then {{math|''X<sup>q</sup>'' − ''X''}} is the product of all monic irreducible polynomials over {{math|GF(''p'')}}, whose degree divides {{math|''n''}}. In fact, if {{math|''P''}} is an irreducible factor over {{math|GF(''p'')}} of {{math|''X<sup>q</sup>'' − ''X''}}, its degree divides {{math|''n''}}, as its [[splitting field]] is contained in {{math|GF(''p''<sup>''n''</sup>)}}. Conversely, if {{math|''P''}} is an irreducible monic polynomial over {{math|GF(''p'')}} of degree {{math|''d''}} dividing {{math|''n''}}, it defines a field extension of degree {{math|''d''}}, which is contained in {{math|GF(''p''<sup>''n''</sup>)}}, and all roots of {{math|''P''}} belong to {{math|GF(''p''<sup>''n''</sup>)}}, and are roots of {{math|''X<sup>q</sup>'' − ''X''}}; thus {{math|''P''}} divides {{math|''X<sup>q</sup>'' − ''X''}}. As {{math|''X<sup>q</sup>'' − ''X''}} does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.

This property is used to compute the product of the irreducible factors of each degree of polynomials over {{math|GF(''p'')}}; see ''[[Distinct degree factorization]]''.