Cyclotomic Polynomials

_n: primitive roots of unity, degree (n).

The idea

The cyclotomic polynomials describe how $x^{n} - 1$ factors over the rational numbers.

The $n$th roots of unity are the $n$ complex solutions of $x^{n} = 1$, so over $\mathbb{C}$ the polynomial $x^{n} - 1$ splits into $n$ linear factors. Over $\mathbb{Q}$ it does not split completely, and to factor it we sort the roots. A root $\zeta$ may satisfy $\zeta^{d} = 1$ for some $d < n$, in which case it is also a $d$th root of unity. Every root has a smallest exponent $d$ with $\zeta^{d} = 1$, its order, and the order divides $n$, so each of the $n$ roots belongs to exactly one divisor $d$ of $n$. A root of order $n$ itself is called primitive. On the unit circle, the eighth roots of unity, for instance, fall into one class for each divisor $d$ of $8$.

Collecting one linear factor per primitive root defines the $n$th cyclotomic polynomial $\Phi_n(x) = \prod_{\zeta \text{ primitive}} (x - \zeta),$ the monic polynomial (leading coefficient $1$) whose roots are exactly the primitive $n$th roots of unity. Two facts, neither obvious from this definition, make it the right factor: its coefficients are integers, and it factors no further over $\mathbb{Q}$.

Ways to work on it

• Walkthrough. Roots, degree (n), and x^n-1 = _d.
• Practice. Degree of _n.
• Hardest. Prove that _p is irreducible over ℚ.

Not sure where to start? Take the ten-question placement test.