Factoring $x^n-1$ over $\mathbb{C}$

In other words, we shall express $x^n-1$ as a product of irreducible polynomials having coeficients in $\mathbb{C}$.

This is, in fact, very easy. The complex roots of $x^n-1$ are precisely the $n^{th}$ roots of unity: those numbers $z$ for which

$$z^n=1.$$

To determine these, recall that any complex number $z$ may be written in its polar decomposition, $z=re^{i\theta}$, where $r\geq 0$ and $0\geq \theta <2\pi$ are it's ``polar coordinates''. (Moreover, if $r > 0$ then these coordinates are unique.) Then $z^n=1$ implies $r^ne^{in\theta}=1$, which in turn implies $r^n=1$ (so $r=1$ since $r\geq 0$) and $n\theta\in \{0,\pm 2\pi,\pm 4\pi, ...\}$. Substituting these into $z=re^{i\theta}$, we find that $z$ must be equal to one of the following numbers:

$$\zeta_n^0=1,\ \ \zeta_n=e^{2\pi i/n}, \ \ \zeta_n^2=e^{4\pi i/n}, \ \ \zeta_n^3=e^{6\pi i/n}, ...\ .$$

In fact, this sequence is periodic, so only the first $n$ terms are distinct and after that the numbers start repeating themselves. We summarize this as the following result.

Lemma 2.10.5   The complex roots of $x^n-1=0$ are

$$\zeta_n^0=1,\ \ \zeta_n=e^{2\pi i/n}, \ \ \zeta_n^2=e^{4\pi i/n}, ..., \zeta_n^{n-1}=e^{2(n-1)\pi i/n}.$$

These may be visualized as follows. In general, plot the complex number

$$z=x+iy=re^{i\theta}=r\cos(\theta)+ir\sin(\theta)$$

at the point $(x,y)=(r\cos(\theta),r\sin(\theta))$ in the $xy$-plane. (Sometimes this way of plotting complex numbers is called the Gaussian plane.) The $n$ roots of $x^n-1=0$ are equally spaced points on the unit circle, starting at $z=1$.

In fact,

$$x^n-1=(x-\zeta_n^0)(x-\zeta_n^1)...(x-\zeta_n^{n-1}) =\prod_{j=0}^{n-1}(x-\zeta_n^j)$$

is the factorization into irreducibles opver $\mathbb{C}$. In other words, factoring $x^n-1$ over $\mathbb{C}$ amounts geometrically to subdividing the circle into $n$ equal parts.

Remark 2.10.6   The ancient Greeks asked for which $n$ is it possible to subdivide the circle into $n$ equal parts using only a ruler and compass. This, and its connection with factoring polynomials, is discussed in further detail in chapter 22 of Schroeder [Sch].