# Primitive Roots of Unity

**Primitive \(n\)th roots of unity** are roots of unity whose multiplicative order is \( n.\) They are the roots of the \(n\)th cyclotomic polynomial, and are central in many branches of number theory, especially algebraic number theory.

## Definition

Let \(n\) be a positive integer. A

primitive \(\ n\)th root of unityis an \(n\)th root of unity that is not a \(k\)th root of unity for any positive \( k<n.\) That is, \( \zeta \) is a primitive \(n\)th root of unity if and only if \[ \zeta^n=1, \text{and } \zeta^k \ne 1 \text{ for any positive integer } k < n. \]

There are four \( 4\)th roots of unity, given by \( \pm 1, \pm i.\) Two of these, namely \( \pm i,\) are primitive. The other two are not: \( 1^1 = 1\) and \( (-1)^2=1.\)

## Basic properties

The primitive \(n\)th roots of unity are the complex numbers \[ e^{2\pi i k/n} : 1\le k \le n, \text{gcd}(k,n)=1. \] There are \( \phi(n)\) primitive \(n\)th roots of unity, where \( \phi(n)\) is Euler's totient function.

Let \( \zeta_n = e^{2\pi i/n}.\) Recall that the \(n\)th roots of unity are the \(n\) distinct powers \( \zeta_n^k = e^{2\pi i k/n} : 1 \le k \le n.\) So it remains to show that \( \zeta_n^k\) is primitive if and only if \(k\) and \(n\) are coprime.

The key fact is that \( \zeta_n\) is a primitive \(n\)th root of unity, since its first \(n\) powers are distinct. A standard order argument shows that \( \zeta_n^a = 1\) if and only if \( a|n,\) since writing \( n = aq+r,\) \( 0 \le r < n,\) gives \(\zeta_n^r = \zeta_n^{n-aq} = 1,\) but this is impossible unless \( r=0.\)

If \( \text{gcd}(k,n) =d,\) then it is easy to check that \( (\zeta_n^k)^{n/d} = 1,\) so if \( d>1 \) then \( \zeta_n^k\) is not primitive. (In fact, it is not hard to show that \( \zeta_n^k\) is a primitive \((n/d)\)th root of unity.)

If \( \text{gcd}(k,n)=1,\) and \( (\zeta_n^k)^r=1,\) then \(n|kr,\) but \(n\) and \(k\) are coprime so \(n|r.\) This shows that \( \zeta_n^k\) is a primitive \(n\)th root of unity, because the first positive power of \(\zeta_n^k\) that equals \( 1\) is \( (\zeta_n^k)^n.\)

Euler's totient function counts the number of positive integers \(k \le n\) that are coprime to \(n,\) which are precisely the exponents that produce primitive \(n\)th roots of unity, so this is the number of primitive \(n\)th roots of unity.

Classify the \(12\)th roots of unity by their multiplicative order.

Let \( \zeta_{12} = e^{2\pi i/12}.\) Then the powers of \( \zeta_{12}\) are classified according to the gcd of the exponent and \( 12:\) as in the above proof, \( \zeta_{12}^k\) is a \( (12/\text{gcd}(12,k))\)th root of unity.

\( \zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11} \) are primitive \( 12\)th roots of unity. (Note \( \phi(12)=4.)\)

\( \zeta_{12}^2, \zeta_{12}^{10} \) are primitive \(6\)th roots of unity.

\( \zeta_{12}^3, \zeta_{12}^9\) are primitive \(4\)th roots of unity.

\( \zeta_{12}^4, \zeta_{12}^8\) are primitive \(3\)rd roots of unity.

\( \zeta_{12}^6 = -1\) is a primitive \(2\)nd root of unity.

\(\zeta_{12}^{12} = 1 \) is a primitive \(1\)st root of unity.

Note that the orders are the divisors of \( 12.\) There are \( \phi(d)\) primitive roots of order \( d,\) for each \( d|12.\) Since \( \sum\limits_{d|12} \phi(d) = 12,\) this accounts for all of the \(12\)th roots of unity.

The above arguments show that the powers of a primitive \(n\)th root of unity enumerate all the primitive \(d\)th roots of unity, for all the divisors \(d\) of \(n.\)

Let \( \zeta_m\) be a primitive \(m^\text{th}\) root of unity, and

let \( \zeta_n\) be a primitive \( n^\text{th}\) root of unity.

Then \( \zeta_m\zeta_n\) is a primitive \(\ell^\text{th}\) root of unity for some positive integer \( \ell.\)

What can we say about \( \ell\) in general?

**Clarification**:

In the answer choices, \(\gcd(\cdot) \) and \(\text{lcm}(\cdot) \) denotes the greatest common divisor function and the lowest common multiple function.

## Sum and product

The product of the primitive \(n\)th roots of unity is \( 1\) unless \( n=2.\) This is because the set of primitive \(n\)th roots of unity, \(n\ge 3,\) can be partitioned into pairs \( \{ \zeta^k, \zeta^{n-k} \}, \) which multiply to give \( 1.\) (For \( n=2\) this fails because \( \zeta^1 \) and \( \zeta^{2-1} \) coincide.)

The sum of the primitive \(n\)th roots of unity is \( \mu(n),\) where \( \mu\) is the Möbius function; see that wiki for a proof.

In fact, there is an elegant formula for the sum of the \(k\)th powers of the primitive \(n\)th roots of unity:

The sum of the \(k\)th powers of the primitive \(n\)th roots of unity is \[ \mu(r) \frac{\phi(n)}{\phi(r)}, \] where \( r = \frac{n}{\text{gcd}(n,k)}.\)

The extremal examples of the theorem are when \( \text{gcd}(n,k) =1\) and when \( \text{gcd}(n,k)=n.\)

When \( \text{gcd}(n,k)=1,\) taking \(k\)th powers permutes the primitive \(n\)th roots of unity, so the sum should still be \( \mu(n).\) Indeed, \( r=n,\) so \( \mu(r)\frac{\phi(n)}{\phi(r)} = \mu(n).\)

When \( \text{gcd}(n,k)=n,\) the powers are all \( 1,\) so the sum is \( \phi(n).\) In this case \( r=1,\) so \( \mu(r)\frac{\phi(n)}{\phi(r)} = \phi(n)\) as desired.

Here is an outline: the sum \( \sum \zeta^k \) is a sum of primitive \(r\)th roots of unity, and it runs over all of them. But there are repetitions: the sum has \(\phi(n)\) terms and there are \(\phi(r)\) primitive \(r\)th roots of unity, so they are each counted \( \frac{\phi(n)}{\phi(r)}\) times. The sum of the primitive \(r\)th roots of unity is \(\mu(r),\) so the result follows.

## Group structure

The \(n\)th roots of unity form a cyclic group under multiplication, generated by \( \zeta_n = e^{2\pi i/n}.\) This group is isomorphic to the additive group \({\mathbb Z}/n\) of the integers mod \(n,\) under the isomorphism \( \zeta_n^k \mapsto k \pmod n.\) In this context, the primitive \(n\)th roots of unity correspond via this isomorphism to the other generators of the group. This is the set of multiplicative units \( ({\mathbb Z}/n)^*\) (which has \(\phi(n)\) elements).

The \(12\)th roots of unity are isomorphic to \( {\mathbb Z}/12.\) The primitive \(12\)th roots of unity, \( \zeta_{12}, \zeta_{12}^5, \zeta_{12}^7,\) and \(\zeta_{12}^{11},\) correspond to the elements \(1,5,7,11\) in \({\mathbb Z}/12.\) These are the elements which generate \( {\mathbb Z}/12 \) additively. E.g. the multiples of \(5\) are \[ 5,10,3,8,1,6,11,4,9,2,7,0, \, 5,10,\ldots \] Every element in \({\mathbb Z}/12\) is on that list.

## Cyclotomic polynomials

The polynomial \[ \prod_{\zeta \text{ a primitive } n\text{th root of unity}} (x-\zeta) \] is a polynomial in \(x\) known as the \(n\)th cyclotomic polynomial. It is of great interest in algebraic number theory. For more details and properties, see the wiki on cyclotomic polynomials.

**Cite as:**Primitive Roots of Unity.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/primitive-roots-of-unity/