Root of unity

An $N$th root of unity is a complex number $z$ satisfying $z^N=1$. The primitive root $e^{2\pi i/N}$ generates all $N$th roots.

A square root of unity is an element $x$ of a ring with identity satisfying $x^2=1$. In a field of characteristic different from two, the only possibilities are $1$ and $-1$: the difference of squares identity $(x-1)(x+1)=0$ has no other roots because a field has no zero divisors. In a general ring more solutions may occur; modulo eight the four residue classes $1,3,5,7$ all have square one.

Let $S$ be a finite multiset of nonzero complex numbers such that taking the $m^2$th powers permutes it for every positive integer $m$. Taking $m=2$ and iterating the resulting permutation shows that each $z\in S$ obeys $z^{4^r}=z$ for some $r\ge 1$, hence is a root of unity. Let $d$ be a common multiple of their orders. The $d^2$th powers are all one, and the assumed permutation then forces every original element to be one. For a unitary matrix, the spectral theorem for normal operators further forces the matrix itself to be the identity.

These roots are taken in an algebraic closure, not merely in ${\mathbb{F}}_q$. If the characteristic of a field is $p$ and $n=p^{a}m$ with $p\nmid m$, then $X^n-1=(X^m-1{)}^{p^a}$; hence there are $m$ distinct roots. They form a cyclic group by the finite multiplicative subgroup of a field is cyclic theorem. In particular they have order $n$ when $\gcd (n,q)=1$.

With $m$ the characteristic-prime-free part of $n$, all roots of unity over a finite field lie in ${\mathbb{F}}_{q^s}$ exactly when $m\mid q^s-1$. This follows from the cyclicity of the multiplicative group of a finite field. For $m=1$ the least positive degree is one. A primitive root of order $m$ has the same field degree, because its Frobenius orbit closes exactly when $m\mid q^s-1$.

The Ramanujan sum is $c_n(k)={\sum }_{1\le j\le n,(j,n)=1}{e}^{2\pi ijk/n}$. It is an integer: insert $1_{(j,n)=1}={\sum }_{d\mid (j,n)}\mu (d)$, sum the geometric series, and obtain $c_n(k)={\sum }_{r\mid (n,k)}r\mu (n/r)$. Here $\mu$ is the Möbius function. Such sums connect roots of unity with integer character of a representation values.

A primitive $n$th root of unity has multiplicative order exactly $n$.

For $\omega =e^{2\pi i/N}$,This finite geometric sum is the orthogonality relation underlying the discrete Fourier transform.

For $\omega =e^{2\pi i/N}$,This is the finite geometric series identity and expresses orthogonality of the characters of ${\mathbb{Z}}_N$.