Root of unity modulo n

In number theory, a kth root of unity modulo n for positive integers k, n Ã¢Â‰Â¥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) . If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. See modular arithmetic for notation and terminology.

The roots of unity modulo are exactly the integers that are coprime with . In fact, these integers are roots of unity modulo by Euler's theorem, and the other integers cannot be roots of unity modulo , because they are zero divisors modulo .

A primitive root modulo, is a generator of the group of units of the ring of integers modulo . There exist primitive roots modulo if and only if where and are respectively the Carmichael function and Euler's totient function.

A root of unity modulo is a primitive th root of unity modulo for some divisor of and, conversely, there are primitive th roots of unity modulo if and only if is a divisor of

Roots of unity

Properties

• If x is a kth root of unity modulo n, then x is a unit (invertible) whose inverse is . That is, x and n are coprime.
• If x is a unit, then it is a (primitive) kth root of unity modulo n, where k is the multiplicative order of x modulo n.
• If x is a kth root of unity and is not a zero divisor, then , because

:

Number of kth roots

For the lack of a widely accepted symbol, we denote the number of kth roots of unity modulo n by . It satisfies a number of properties:

• for
• where λ denotes the Carmichael function and denotes Euler's totient function
• is a multiplicative function
• where the bar denotes divisibility
• where denotes the least common multiple
• For prime , . The precise mapping from to is not known. If it were known, then together with the previous law it would yield a way to evaluate quickly.

Examples

Let and . In this case, there are three cube roots of unity (1, 2, and 4). When 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 kth roots.

Primitive roots of unity

Properties

• The maximum possible radix exponent for primitive roots modulo is , where λ denotes the Carmichael function.
• A radix exponent for a primitive root of unity is a divisor of .
• Every divisor of yields a primitive th root of unity. One can obtain such a root by choosing a th primitive root of unity (that must exist by definition of λ), named and compute the power .
• If x is a primitive kth 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 . Since k is minimal, it must be and is a divisor of Ã¢Â„“.

Number of primitive kth roots

For the lack of a widely accepted symbol, we denote the number of primitive kth roots of unity modulo n by . It satisfies the following properties:

• Consequently the function has values different from zero, where computes the number of divisors.
• for , since -1 is always a square root of 1.
• for
• for and in
• with being Euler's totient function
• The connection between and can be written in an elegant way using a Dirichlet convolution:

: , i.e.

One can compute values of recursively from using this formula, which is equivalent to the Möbius inversion formula.

Testing whether x is a primitive kth root of unity modulo n

By fast exponentiation, one can check that . If this is true, x is a kth 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 . In order to exclude this possibility, one has only to check for a few ℓ's equal k divided by a prime.

That is, x is a primitive kth root of unity modulo n if and only if and

for every prime divisor p of n.

For example, if every positive integer less than 17 is a 16th root of unity modulo 17, and the integers that are primitive 16th roots of unity modulo 17 are exactly those such that

Finding a primitive kth root of unity modulo n

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

Finding multiple primitive kth roots modulo n

Once a primitive kth root of unity x is obtained, every power is a th root of unity, but not necessarily a primitive one. The power is a primitive th root of unity if and only if and are coprime. The proof is as follows: If is not primitive, then there exists a divisor of with , and since and are coprime, there exists integers such that . This yields

,

which means that is not a primitive th root of unity because there is the smaller exponent .

That is, by exponentiating x one can obtain different primitive kth 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 kth root of unity modulo n

In what integer residue class rings does a primitive kth root of unity exist? It can be used to compute a discrete Fourier transform (more precisely a number theoretic transform) of a -dimensional integer vector. In order to perform the inverse transform, divide by ; that is, k is also a unit modulo

A simple way to find such an n is to check for primitive kth roots with respect to the moduli in the arithmetic progression 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 , it holds . Thus if is prime, then , and thus there are primitive kth roots of unity. But the test for primes is too strong, and there may be other appropriate moduli.

Finding an n with multiple primitive roots of unity modulo n

To find a modulus such that there are primitive roots of unity modulo , the following theorem reduces the problem to a simpler one:

For given there are primitive roots of unity modulo if and only if there is a primitive th root of unity modulo n.

Proof

Backward direction: If there is a primitive th root of unity modulo called , then is a th root of unity modulo .

Forward direction: If there are primitive roots of unity modulo , then all exponents are divisors of . This implies and this in turn means there is a primitive th root of unity modulo .

References