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In [[mathematics]], a '''''k''-th root of unity modulo ''n''''' for positive [[integer]]s ''k'', ''n''&nbsp;≥&nbsp;2, is a solution ''x'' to the [[congruence]] <math>x^k \equiv 1 \pmod{n} </math>. If ''k'' is the smallest such exponent for ''x'', then ''x'' is called a '''primitive ''k''-th root of unity modulo ''n'''''.<ref>
{{Cite journal
  | last1 = Finch
  | first1 = Stephen
  | last2 = Martin
  | first2 = Greg
  | last3 = Sebah
  | first3 = And Pascal
  | title = Roots of unity and nullity modulo ''n''
  | journal = Proceedings of the American Mathematical Society
  | volume = 138
  | issue = 8
  | pages = 2729–2743
  | publisher = American Mathematical Society
  | url = http://www.math.ubc.ca/~gerg/papers/downloads/RUNM.pdf
  | accessdate = 2011-02-20}}
</ref>
 
Do not confuse this with a [[Primitive root modulo n|primitive element modulo ''n'']], where the primitive element must generate all [[Unit (ring theory)|units]] of the [[residue class ring]] <math>\mathbb{Z}/n\mathbb{Z}</math> by exponentiation.
That is, there are always roots and primitive roots of unity modulo ''n'' for ''n''&nbsp;≥&nbsp;2, but for some ''n'' there is no primitive element modulo ''n''. Being a root or a primitive root modulo ''n'' always depends on the exponent ''k'' and the modulus ''n'', whereas being a primitive element modulo ''n'' only depends on the modulus ''n'' — the exponent is automatically <math>\varphi(n)</math>.
 
== Roots of unity ==
 
=== Properties ===
 
* If ''x'' is a ''k''-th root of unity, then ''x'' is a unit (invertible) whose inverse is <math>x^{k-1}</math>. 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 <math>x-1</math> is not a [[zero divisor]], then <math>\sum_{j=0}^{k-1} x^j \equiv 0 \pmod{n}</math>, because
:: <math>(x-1)\cdot\sum_{j=0}^{k-1} x^j \equiv x^k-1 \equiv 0 \pmod{n}.</math>
 
=== 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 <math>f(n,k)</math>.
It satisfies a number of properties:
 
* <math>f(n,1) = 1</math> for <math>n\ge2</math>
* <math>f(n,\lambda(n)) = \varphi(n)</math> where &lambda; denotes the [[Carmichael function]] and <math>\varphi</math> denotes [[Euler's totient function]]
* <math>n \mapsto f(n,k)</math> is a [[multiplicative function]]
* <math>k\mid l \implies f(n,k)\mid f(n,l)</math> where the bar denotes [[divisibility]]
* <math>f(n,\mathrm{lcm}(a,b)) = \mathrm{lcm}(f(n,a),f(n,b))</math> where <math>\mathrm{lcm}</math> denotes the [[least common multiple]]
* For [[prime]] <math>p</math>, <math>\forall i\in\mathbb{N}\ \exists j\in\mathbb{N}\ f(n,p^i) = p^j</math>. The precise mapping from <math>i</math> to <math>j</math> is not known. If it would be known, then together with the previous law it would yield a way to evaluate <math>f</math> quickly.
 
== Primitive roots of unity ==
 
=== Properties ===
 
* The maximum possible radix exponent for primitive roots modulo <math>n</math> is <math>\lambda(n)</math>, where &lambda; denotes the [[Carmichael function]].
* A radix exponent for a primitive root of unity is a [[divisor]] of <math>\lambda(n)</math>.
* Every divisor <math>k</math> of <math>\lambda(n)</math> yields a primitive <math>k</math>-th root of unity. You can obtain one by choosing a <math>\lambda(n)</math>-th primitive root of unity (that must exist by definition of &lambda;), named <math>x</math> and compute the power <math>x^{\lambda(n)/k}</math>.
* If ''x'' is a primitive ''k''-th root of unity and also a (not necessarily primitive) ℓ-th 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 <math>\mathrm{gcd}(k,\ell)</math>. Since ''k'' is minimal, it must be <math>k = \mathrm{gcd}(k,\ell)</math> and <math>\mathrm{gcd}(k,\ell)</math> is a divisor of ℓ.
<!--
* If <math>x</math> is a primitive <math>k</math>-th root of unity and for another exponent <math>l</math> it holds <math>x^l \equiv 1 \pmod{n}</math>, then <math>k</math> is a divisor of <math>l</math>.
-->
 
=== 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 <math>g(n,k)</math>.
It satisfies the following properties:
 
* <math>g(n,k) = \begin{cases} >0 &\text{if } k\mid\lambda(n), \\ 0 & \text{otherwise}. \end{cases}</math>
* Consequently the function <math>k \mapsto g(n,k)</math> has <math>d(\lambda(n))</math> values different from zero, where <math>d</math> computes the [[number of divisors]].
* <math>g(n,1) = 1</math>
* <math>g(4,2) = 1</math>
* <math>g(2^n,2) = 3</math> for <math>n \ge 3</math>, since -1 is always a [[square root]] of 1.
* <math>g(2^n,2^k) = 2^k</math> for <math>k \in [2,n-1)</math>
* <math>g(n,2) = 1</math> for <math>n \ge 3</math> and <math>n</math> in {{OEIS|A033948}}
* <math>\sum_{k\in\mathbb{N}} g(n,k) = f(n,\lambda(n)) = \varphi(n)</math> with <math>\varphi</math> being [[Euler's totient function]]
* The connection between <math>f</math> and <math>g</math> can be written in an elegant way using a [[Dirichlet convolution]]:
:: <math>f = \bold{1} * g</math>, i.e. <math>f(n,k) = \sum_{d\mid k} g(n,d)</math>
: You can compute values of <math>g</math> recursively from <math>f</math> using this formula, which is equivalent to the [[Möbius inversion formula]].
 
=== Testing whether ''x'' is a primitive ''k''-th root of unity modulo ''n'' ===
 
By [[Exponentiation by squaring|fast exponentiation]] you can check that <math>x^k \equiv 1 \pmod{n}</math>.
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 <math>x^{\ell} \equiv 1 \pmod{n}</math>.
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:
:<math>\forall p \text{ prime dividing}\ k,\quad x^{k/p} \not\equiv 1 \pmod{n}.</math>
 
=== Finding a primitive ''k''-th root of unity modulo ''n'' ===
 
Among the primitive ''k''-th roots of unity, the primitive <math>\lambda(n)</math>-th roots are most frequent.
It is thus recommended to try some integers for being a primitive <math>\lambda(n)</math>-th root, what will succeed quickly.
For a primitive <math>\lambda(n)</math>-th root ''x'', the number <math>x^{\lambda(n)/k}</math> is a primitive <math>k</math>-th root of unity.
If ''k'' does not divide <math>\lambda(n)</math>, 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 <math>x^l</math> is a <math>k</math>-th root of unity, but not necessarily a primitive one. The power <math>x^l</math> is a primitive <math>k</math>-th root of unity if and only if <math>k</math> and <math>l</math> are [[coprime]]. The proof is as follows: If <math>x^l</math> is not primitive, then there exists a divisor <math>m</math> of <math>k</math> with <math>(x^l)^m \equiv 1 \pmod{n}</math>, and since <math>k</math> and <math>l</math> are coprime, there exists an inverse <math>l^{-1}</math> of <math>l</math> modulo <math>k</math>. This yields <math>1 \equiv ((x^l)^m)^{l^{-l}} \equiv x^m \pmod{n}</math>, which means that <math>x</math> is not a primitive <math>k</math>-th root of unity because there is the smaller exponent <math>m</math>.
 
That is, by exponentiating ''x'' one can obtain <math>\varphi(k)</math> 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 ring]]s 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 <math>k</math>-dimensional integer [[Vector (mathematics and physics)|vector]].
In order to perform the inverse transform, you also need to divide by <math>k</math>, that is, ''k'' shall also be a unit modulo <math>n</math>.
 
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]] <math>k+1, 2k+1, 3k+1, \dots</math>.
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 <math>p</math> it holds <math>\lambda(p) = p-1</math>.
Thus if <math>mk+1</math> is prime then <math>\lambda(mk+1) = mk</math> 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 <math>n</math> such that there are primitive <math>k_1</math>-th, <math>k_2</math>-th, ..., <math>k_m</math>-th roots of unity modulo <math>n</math>, the following theorem reduces the problem to a simpler one:
 
: For given <math>n</math> there are primitive <math>k_1</math>-th, ..., <math>k_m</math>-th roots of unity modulo <math>n</math> if and only if there is a primitive <math>\mathrm{lcm}(k_1, ..., k_m)</math>-th root of unity modulo ''n''.
 
; Proof
 
Backward direction:
If there is a primitive <math>\mathrm{lcm}(k_1, ..., k_m)</math>-th root of unity modulo <math>n</math> called <math>x</math>, then <math>x^{\mathrm{lcm}(k_1, \dots, k_m)/k_l}</math> is a <math>k_l</math>-th root of unity modulo <math>n</math>.
 
Forward direction:
If there are primitive <math>k_1</math>-th, ..., <math>k_m</math>-th roots of unity modulo <math>n</math>, then all exponents <math>k_1, \dots, k_m</math> are divisors of <math>\lambda(n)</math>. This implies <math>\mathrm{lcm}(k_1, \dots, k_m) \mid \lambda(n)</math> and this in turn means there is a primitive <math>\mathrm{lcm}(k_1, ..., k_m)</math>-th root of unity modulo <math>n</math>.
 
== See also ==
* [[Root of unity]]
* [[Primitive root modulo n|Primitive root modulo ''n'']]
 
== References ==
{{reflist}}
 
[[Category:Modular arithmetic]]

Revision as of 11:22, 1 March 2014

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