Root Of Unity Modulo N

In mathematics, namely ring theory, a ''k''th root of unity modulo ''n'' for positive integers ''k'', ''n'' ≥ 2, is a root of unity in the ring of integers modulo ''n'', that is, a solution ''x'' to the equation (or ''congruence'') $x^k \equiv 1 \pmod$. If ''k'' is the smallest such exponent for ''x'', then ''x'' is called a primitive ''k''th root of unity modulo ''n''.

See modular arithmetic for notation and terminology.

Software to compute modular roots is available in the python library sympy.ntheory.residue_ntheory using the function nthroot_mod. See https://docs.sympy.org/latest/modules/ntheory.html.

Do not confuse this with a primitive root modulo ''n'', which is a generator of the group of units of the ring of integers modulo ''n''. The primitive roots modulo ''n'' are the primitive $\varphi(n)$-roots of unity modulo ''n'', where $\varphi$ is Euler's totient function.

This topic deals with roots of unity modulo ''n'' where ''n'' is at times not a prime number. addresses the special case where ''n'' is prime and Root of unity covers (non-modular) roots of unity in the field of complex numbers.

Roots of unity

Properties

* If ''x'' is a ''k''th root of unity modulo ''n'', then ''x'' is a unit (invertible) whose inverse is $x^$. That is, ''x'' and ''n'' are coprime.
* If ''x'' is a unit, then it is a (primitive) ''k''th root of unity modulo ''n'', where ''k'' is the multiplicative order of ''x'' modulo ''n''.
* If ''x'' is a ''k''th root of unity and $x-1$ is not a zero divisor, then $\sum_^ x^j \equiv 0 \pmod$, because
:: $(x-1)\cdot\sum_^ x^j \equiv x^k-1 \equiv 0 \pmod.$

Number of ''k''th roots

For the lack of a widely accepted symbol, we denote the number of ''k''th roots of unity modulo ''n'' by $f(n,k)$.
It satisfies a number of properties:

* $f(n,1) = 1$ for $n\ge2$
* $f(n,\lambda(n)) = \varphi(n)$ where λ denotes the Carmichael function and $\varphi$ denotes Euler's totient function
* $n \mapsto f(n,k)$ is a multiplicative function
* $k\mid l \implies f(n,k)\mid f(n,l)$ where the bar denotes divisibility
* $f(n,\mathrm(a,b)) = \mathrm(f(n,a),f(n,b))$ where $\mathrm$ denotes the least common multiple
* For prime $p$, $\forall i\in\mathbb\ \exists j\in\mathbb\ f(n,p^i) = p^j$. The precise mapping from $i$ to $j$ is not known. If it were known, then together with the previous law it would yield a way to evaluate $f$ quickly.

Examples

Let $n = 7$ and $k = 3$. In this case, there are three cube roots of unity (1, 2, and 4). When $n = 11$ however, there is only one cube root of unity, the unit 1 itself. This behavior is quite different from the field of complex numbers where every nonzero number has ''k'' ''k''th roots.

Primitive roots of unity

Properties

* The maximum possible radix exponent for primitive roots modulo $n$ is $\lambda(n)$, where λ denotes the Carmichael function.
* A radix exponent for a primitive root of unity is a divisor of $\lambda(n)$.
* Every divisor $k$ of $\lambda(n)$ yields a primitive $k$th root of unity. You can obtain one by choosing a $\lambda(n)$th primitive root of unity (that must exist by definition of λ), named $x$ and compute the power $x^$.
* If ''x'' is a primitive ''k''th root of unity and also a (not necessarily primitive) ℓth root of unity, then ''k'' is a divisor of ℓ. This is true, because Bézout's identity yields an integer linear combination of ''k'' and ℓ equal to $\mathrm(k,\ell)$. Since ''k'' is minimal, it must be $k = \mathrm(k,\ell)$ and $\mathrm(k,\ell)$ is a divisor of ℓ.

Number of primitive ''k''th roots

For the lack of a widely accepted symbol, we denote the number of primitive ''k''th roots of unity modulo ''n'' by $g(n,k)$.
It satisfies the following properties:

* $g(n,k) = \begin >0 &\text k\mid\lambda(n), \\ 0 & \text. \end$
* Consequently the function $k \mapsto g(n,k)$ has $d(\lambda(n))$ values different from zero, where $d$ computes the number of divisors.
* $g(n,1) = 1$
* $g(4,2) = 1$
* $g(2^n,2) = 3$ for $n \ge 3$, since -1 is always a square root of 1.
* $g(2^n,2^k) = 2^k$ for $k \in [2,n-1)$
* $g(n,2) = 1$ for $n \ge 3$ and $n$ in
* $\sum_ g(n,k) = f(n,\lambda(n)) = \varphi(n)$ with $\varphi$ being Euler's totient function
* The connection between $f$ and $g$ can be written in an elegant way using a Dirichlet convolution:
:: $f = \mathbf * g$, i.e. $f(n,k) = \sum_ g(n,d)$
: You can compute values of $g$ recursively from $f$ using this formula, which is equivalent to the [ öbius inversion formula.

Testing whether ''x'' is a primitive ''k''th root of unity modulo ''n''

By fast exponentiation you can check that $x^k \equiv 1 \pmod$. If this is true, ''x'' is a ''k''th root of unity modulo ''n'' but not necessarily primitive. If it is not a primitive root, then there would be some divisor ℓ of ''k'', with $x^ \equiv 1 \pmod$. In order to exclude this possibility, one has only to check for a few ℓ's equal ''k'' divided by a prime. That is, what needs to be checked is:

:$\forall p \text\ k,\quad x^ \not\equiv 1 \pmod.$

Finding a primitive ''k''th root of unity modulo ''n''

Among the primitive ''k''th roots of unity, the primitive $\lambda(n)$th roots are most frequent.
It is thus recommended to try some integers for being a primitive $\lambda(n)$th root, what will succeed quickly.
For a primitive $\lambda(n)$th root ''x'', the number $x^$ is a primitive $k$th root of unity.
If ''k'' does not divide $\lambda(n)$, then there will be no ''k''th roots of unity, at all.

Finding multiple primitive ''k''th roots modulo ''n''

Once you have a primitive ''k''th root of unity ''x'', every power $x^l$ is a $k$th root of unity, but not necessarily a primitive one. The power $x^l$ is a primitive $k$th root of unity if and only if $k$ and $l$ are coprime. The proof is as follows: If $x^l$ is not primitive, then there exists a divisor $m$ of $k$ with $(x^l)^m \equiv 1 \pmod$, and since $k$ and $l$ are coprime, there exists an inverse $l^$ of $l$ modulo $k$. This yields $1 \equiv ((x^l)^m)^ \equiv x^m \pmod$, which means that $x$ is not a primitive $k$th root of unity because there is the smaller exponent $m$.

That is, by exponentiating ''x'' one can obtain $\varphi(k)$ different primitive ''k''th roots of unity, but these may not be all such roots. However, finding all of them is not so easy.

Finding an ''n'' with a primitive ''k''th root of unity modulo ''n''

You may want to know, in what integer residue class rings you have a primitive ''k''th root of unity. You need it for instance if you want to compute a Discrete Fourier Transform (more precisely a Number theoretic transform) of a $k$-dimensional integer vector. In order to perform the inverse transform, you also need to divide by $k$, that is, ''k'' shall also be a unit modulo $n.$

A simple way to find such an ''n'' is to check for primitive ''k''th roots with respect to the moduli in the arithmetic progression $k+1, 2k+1, 3k+1, \dots$. All of these moduli are coprime to ''k'' and thus ''k'' is a unit. According to Dirichlet's theorem on arithmetic progressions there are infinitely many primes in the progression, and for a prime $p$ it holds $\lambda(p) = p-1$. Thus if $mk+1$ is prime then $\lambda(mk+1) = mk$ and thus you have primitive ''k''th roots of unity. But the test for primes is too strong, you may find other appropriate moduli.

Finding an ''n'' with multiple primitive roots of unity modulo ''n''

If you want to have a modulus $n$ such that there are primitive $k_1$th, $k_2$th, ..., $k_m$th roots of unity modulo $n$, the following theorem reduces the problem to a simpler one:

: For given $n$ there are primitive $k_1$th, ..., $k_m$th roots of unity modulo $n$ if and only if there is a primitive $\mathrm(k_1, ..., k_m)$th root of unity modulo ''n''.

; Proof

Backward direction:
If there is a primitive $\mathrm(k_1, ..., k_m)$th root of unity modulo $n$ called $x$, then $x^$ is a $k_l$th root of unity modulo $n$.

Forward direction:
If there are primitive $k_1$th, ..., $k_m$th roots of unity modulo $n$, then all exponents $k_1, \dots, k_m$ are divisors of $\lambda(n)$. This implies $\mathrm(k_1, \dots, k_m) \mid \lambda(n)$ and this in turn means there is a primitive $\mathrm(k_1, ..., k_m)$th root of unity modulo $n$.

References

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Modular arithmetic