Root of Unity - Elementary Facts

Elementary Facts

Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n: if z1 = 1 then z is a primitive first root of unity, otherwise if z2 = 1 then z is a primitive second (square) root of unity, otherwise, ..., and by assumption there must be a "1" at or before the nth term in the sequence.

If z is an nth root of unity and a ≡ b (mod n) then za = zb. By the definition of congruence, a = b + kn for some integer k. But then,

Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. This is often convenient.

Any integer power of an nth root of unity is also an nth root of unity:

Here k may be negative. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity:

Let z be a primitive nth root of unity. Then the powers z, z2, ... zn−1, zn = z0 = 1 are all distinct. Assume the contrary, that za = zb where 1 ≤ a < b ≤ n. Then zb−a = 1. But 0 < b−a < n, which contradicts z being primitive.

Since an nth degree polynomial equation can only have n distinct roots, this implies that the powers of a primitive root z, z2, ... zn−1, zn = z0 = 1 are in fact all of the nth roots of unity.

From the preceding facts it follows that if z is a primitive nth root of unity:

If z is not primitive there is only one implication:

An example showing that the converse implication is false is given by:

Let z be a primitive nth root of unity and let k be a positive integer. From the above discussion, zk is a primitive root of unity for some a. Now if zka = 1, ka must be a multiple of n. The smallest number that is divisible by both n and k is their least common multiple, denoted by lcm(n, k). It is related to their greatest common divisor, gcd(n, k), by the formula:

i.e.

Therefore, zk is a primitive ath root of unity where

Thus, if k and n are coprime, zk is also a primitive nth root of unity, and therefore there are φ(n) (where φ is Euler's totient function) distinct primitive nth roots of unity. (This implies that if n is a prime number, all the roots except +1 are primitive).

In other words, if R(n) is the set of all nth roots of unity and P(n) is the set of primitive ones, R(n) is a disjoint union of the P(n):

where the notation means that d goes through all the divisors of n, including 1 and n.

Since the cardinality of R(n) is n, and that of P(n) is φ(n), this demonstrates the classical formula