Editing Cyclotomic polynomial (section)

{{short description|Irreducible polynomial whose roots are nth roots of unity}}
In [[mathematics]], the''' ''n''th cyclotomic polynomial''', for any [[positive integer]] ''n'', is the unique [[irreducible polynomial]] with integer [[Coefficient|coefficients]] that is a [[divisor]] of <math>x^n-1</math> and is not a divisor of <math>x^k-1</math> for any {{nowrap|''k'' < ''n''.}} Its [[root of a function|roots]] are all ''n''th [[primitive root of unity|primitive roots of unity]] 
<math>
e^{2i\pi\frac{k}{n}}
</math>, where ''k'' runs over the positive integers less than ''n'' and [[coprime integers|coprime]] to ''n'' (and ''i'' is the [[imaginary unit]]). In other words, the ''n''th cyclotomic polynomial is equal to
:<math>
\Phi_n(x) =
\prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}
\left(x-e^{2i\pi\frac{k}{n}}\right).
</math>

It may also be defined as the [[monic polynomial]] with integer coefficients that is the [[minimal polynomial (field theory)|minimal polynomial]] over the [[Field (mathematics)|field]] of the [[rational number]]s of any [[Primitive root of unity|primitive ''n''th-root of unity]] (<math> e^{2i\pi/n} </math> is an example of such a root).

An important relation linking cyclotomic polynomials and primitive roots of unity is
:<math>\prod_{d\mid n}\Phi_d(x) = x^n - 1,</math>
showing that <math>x</math> is a root of <math>x^n - 1</math> if and only if it is a ''d''{{space|hair}}th primitive root of unity for some ''d'' that divides ''n''.<ref>{{citation
| last=Roman 
| first=Steven
| title=Advanced Linear Algebra | edition=Third 
| series=[[Graduate Texts in Mathematics]] | publisher  = Springer 
| date=2008| pages= 
| isbn=978-0-387-72828-5 
| at = p. 465 §18
|author-link=Steven Roman}}</ref>