Editing Cyclotomic polynomial (section)

== Cyclotomic polynomials over a finite field and over the {{math|''p''}}-adic integers ==
{{see also|Finite field#Roots of unity}}
Over a [[finite field]] with a prime number {{math|''p''}} of elements, for any integer {{math|''n''}} that is not a multiple of {{math|''p''}}, the cyclotomic polynomial <math>\Phi_n</math> factorizes into <math>\frac{\varphi (n)}{d}</math> irreducible polynomials of degree {{math|''d''}}, where <math>\varphi (n)</math> is [[Euler's totient function]] and {{math|''d''}} is the [[multiplicative order]] of {{math|''p''}} modulo {{math|''n''}}. In particular, <math>\Phi_n</math> is irreducible [[if and only if]] {{math|''p''}} is a [[primitive root modulo n|primitive root modulo {{mvar|n}}]], that is, {{math|''p''}} does not divide {{math|''n''}}, and its multiplicative order modulo {{math|''n''}} is <math>\varphi(n)</math>, the degree of <math>\Phi_n</math>.<ref>{{citation | last1 = Lidl | first1 = Rudolf | last2 = Niederreiter| first2 = Harald | edition = 2nd  |  page = 65 | publisher = Cambridge University Press | title = Finite Fields | year = 2008}}.</ref>

These results are also true over the [[p-adic integer|{{mvar|p}}-adic integers]], since [[Hensel's lemma]] allows lifting a factorization over the field with {{math|''p''}} elements to a factorization over the {{math|''p''}}-adic integers.