Root Of Unity

A root of unity is a complex number that, when raised to some positive integer power n, yields the number 1. Think of it as a number that "comes back to itself" after a specific number of multiplications. These numbers are not just mathematical curiosities; they're fundamental tools in various branches of mathematics, including number theory, the intricate study of group characters, and the powerful discrete Fourier transform. They've even earned a nickname, sometimes called de Moivre numbers, after the French mathematician Abraham de Moivre, which is rather charming, if a bit predictable.

Roots of unity aren't confined to the realm of complex numbers. They can be defined within any field. If the field has a characteristic of zero, then these roots are, predictably, complex numbers that also happen to be algebraic integers. Now, for fields with a positive characteristic, the situation shifts. Here, the roots reside within a finite field. Interestingly, the converse holds true as well: every non-zero element in a finite field is, in fact, a root of unity. And in any algebraically closed field, you'll find exactly n distinct nth roots of unity, unless n happens to be a multiple of the field's positive characteristic.

General Definition

An nth root of unity, where n is a positive integer, is a number z that satisfies the equation:

[1] [2] $z^n = 1$

When we talk about roots of unity without specifying the domain, we usually mean complex numbers. This includes the obvious case of 1 itself, and also -1 if n is even (since -1 is just a complex number with a zero imaginary part). In the complex plane, the nth roots of unity are given by the formula:

[3] $\exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.$

This formula is elegant, but the defining equation $z^n = 1$ is far more general. It holds true not just in fields, but even within any ring. So, we can indeed talk about roots of unity residing in fields other than the complex numbers. As mentioned, if the field's characteristic is zero, these roots will be complex numbers. If the characteristic is positive, they'll be found within a finite field. It’s a neat bit of symmetry, really. For more on roots of unity in fields with positive characteristic, one might delve into the Finite field § Roots of unity section. For those interested in modular arithmetic, the topic of roots of unity modulo n is also covered.

An nth root of unity is termed "primitive" if it's not an mth root of unity for any smaller positive integer m. In other words, it’s the "lowest power" that brings you back to 1. Mathematically, this means:

[4] [5] $z^n = 1 \quad {\text{and}} \quad z^m \neq 1 \quad \text{for } m=1,2,3,\ldots ,n-1.$

A helpful shortcut: if n is a prime number, then all nth roots of unity, except for the trivial root 1, are primitive. [6] Going back to the exponential formula, the primitive nth roots of unity are precisely those where the integer k and n are coprime integers.

The subsequent discussion will focus on complex roots of unity. If you're curious about fields with non-zero characteristic, the Finite field § Roots of unity section is your destination. And for those dealing with modular arithmetic, the "Root of unity modulo n" article will be more relevant.

Elementary Properties

Every nth root of unity z has a smallest positive integer a such that $z^a = 1$ . This a is the "order" of the root, and z is a primitive ath root of unity.

Consider any integer power of an nth root of unity. It turns out to be another nth root of unity. This is quite straightforward:

[7] $(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.$

This property also holds for negative exponents. In fact, the reciprocal of an nth root of unity is its complex conjugate, and it’s also an nth root of unity:

[8] $\frac{1}{z} = z^{-1} = z^{n-1} = \bar{z}.$

Now, let's say you have an nth root of unity z, and you have two integers, a and b, such that $a \equiv b \pmod n$ . This means a and b leave the same remainder when divided by n. In this case, $z^a = z^b$ . Why? Because $a = b + kn$ for some integer k, and then:

$z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b (1)^k = z^b.$

This implies that if you're dealing with $z^a$ , you can simplify it to $z^r$ , where r is the remainder when a is divided by n (so $0 \le r < n$ ).

If z is a primitive nth root of unity, then the sequence of its powers, $z^1, z^2, \dots, z^{n-1}, z^n = 1$ , gives you all n distinct nth roots of unity. If any two powers, say $z^a$ and $z^b$ with $1 \le a < b \le n$ , were equal, then $z^{b-a}$ would be 1. But since $b-a$ is smaller than n, this would contradict the fact that z is primitive. Since an nth-degree polynomial equation over the complex numbers can have at most n roots, we've found them all.

Consequently, if z is a primitive nth root of unity, then $z^a = z^b$ if and only if $a \equiv b \pmod n$ . The converse isn't always true if z isn't primitive. For example, if $n=4$ , $z=-1$ is a non-primitive 4th root of unity. We have $z^2 = (-1)^2 = 1$ and $z^4 = (-1)^4 = 1$ . But $2 \not\equiv 4 \pmod 4$ . So, while congruence implies equality of powers, the reverse isn't guaranteed for non-primitive roots.

Now, let z be a primitive nth root of unity. Consider a power $w = z^k$ . This w will be a primitive ath root of unity, where:

$a = \frac{n}{\gcd(k,n)}$

Here, $\gcd(k,n)$ is the greatest common divisor of k and n. This formula arises because a represents the smallest multiple of k that is also a multiple of n. In other words, a is related to the least common multiple of k and n. Specifically, $ak = \operatorname{lcm}(k,n) = \frac{kn}{\gcd(k,n)}$ , which simplifies to $a = \frac{n}{\gcd(k,n)}$ .

This means that if k and n are coprime, then $z^k$ is also a primitive nth root of unity. Consequently, there are exactly $\phi(n)$ distinct primitive nth roots of unity, where $\phi$ is Euler's totient function. This explains why, if n is a prime number, all roots except 1 are primitive.

To put it another way, let $R(n)$ be the set of all nth roots of unity and $P(n)$ be the set of primitive nth roots. Then $R(n)$ can be expressed as a disjoint union of $P(d)$ for all positive divisors d of n:

$R(n) = \bigcup_{d\,|\,n} P(d)$

Since the cardinality of $R(n)$ is n and the cardinality of $P(n)$ is $\phi(n)$ , this relationship leads directly to the well-known formula:

$\sum_{d\,|\,n} \phi(d) = n$

Group Properties

Group of All Roots of Unity

The set of all roots of unity forms a fascinating mathematical structure. If you take the product of two roots of unity, the result is another root of unity. Similarly, the multiplicative inverse of a root of unity is also a root of unity. Let's say we have $x^m = 1$ and $y^n = 1$ . Then $(x^{-1})^m = 1$ , and $(xy)^k = 1$ , where k is the least common multiple of m and n.

This closure under multiplication and inversion means that the roots of unity form an abelian group under multiplication. This group is particularly significant as it represents the torsion subgroup of the circle group, which consists of all complex numbers with magnitude 1.

Group of nth Roots of Unity

For any given integer n, the set of nth roots of unity also forms a well-behaved structure. The product and multiplicative inverse of any two nth roots of unity are themselves nth roots of unity. This makes the set of nth roots of unity an abelian group under multiplication.

If we pick a primitive nth root of unity, say $\omega$ , then all other nth roots are simply powers of $\omega$ . This tells us that the group of nth roots of unity is a cyclic group. The name "cyclic group" itself originates from this property, as these groups are fundamentally linked to the circle group.

Galois Group of the Primitive nth Roots of Unity

Consider the field extension $\mathbb{Q}(\omega)$ of the rational numbers $\mathbb{Q}$ , generated by a primitive nth root of unity $\omega$ . Since all nth roots of unity are powers of $\omega$ , this field $\mathbb{Q}(\omega)$ contains every nth root of unity. Moreover, it's a Galois extension of $\mathbb{Q}$ .

An integer k generates a primitive nth root of unity $\omega^k$ if and only if k and n are coprime. The mapping $\omega \mapsto \omega^k$ induces an automorphism of $\mathbb{Q}(\omega)$ . This automorphism acts on every nth root of unity by raising it to the kth power. Every automorphism of $\mathbb{Q}(\omega)$ can be represented in this manner. These automorphisms collectively form the Galois group of $\mathbb{Q}(\omega)$ over the rationals.

The rules of exponentiation reveal that the composition of two such automorphisms corresponds to multiplying their exponents. This leads to a group isomorphism between the units of the ring of integers modulo n (denoted $(\mathbb{Z}/n\mathbb{Z})^\times$ ) and the Galois group of $\mathbb{Q}(\omega)$ over $\mathbb{Q}$ . This abelian nature of the Galois group is crucial, as it implies that primitive nth roots of unity can be expressed using radicals.

Galois Group of the Real Part of the Primitive nth Roots of Unity

The real parts of the primitive nth roots of unity are intricately connected. They are the roots of the minimal polynomial of $2\cos(2\pi/n)$ . These roots themselves form a cyclic Galois group.

Trigonometric Expression

De Moivre's formula, a cornerstone of complex number manipulation, states that for any real number x and integer n:

$(\cos x + i \sin x)^n = \cos nx + i \sin nx$

If we set $x = \frac{2\pi}{n}$ , we get a primitive nth root of unity:

$\left(\cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}\right)^n = \cos 2\pi + i \sin 2\pi = 1$

Crucially, for $k = 1, 2, \dots, n-1$ , the kth power of this expression is not equal to 1:

$\left(\cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}\right)^k = \cos \frac{2k\pi}{n} + i \sin \frac{2k\pi}{n} \neq 1$

This means that $\cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}$ is indeed a primitive nth root of unity.

This trigonometric representation offers a beautiful geometric interpretation. In the complex plane, the nth roots of unity are located at the vertices of a regular n-sided polygon inscribed within the unit circle, with one vertex fixed at the number 1. This geometric insight is the origin of the term "cyclotomic," derived from the Greek words for "circle" and "cut," appearing in phrases like cyclotomic field and cyclotomic polynomial.

Euler's formula, $e^{ix} = \cos x + i \sin x$ , provides an even more concise way to express the nth roots of unity:

$e^{2\pi i \frac{k}{n}}, \quad 0 \le k < n$

As established earlier, these are primitive nth roots if and only if the fraction $k/n$ is in its lowest terms, meaning k and n are coprime. Numbers that can be expressed as the real part of a root of unity, like $\cos(2\pi k/n)$ , are known as trigonometric numbers.

Algebraic Expression

By their very definition, the nth roots of unity are the roots of the polynomial $x^n - 1$ . This places them firmly in the category of algebraic numbers. However, the polynomial $x^n - 1$ is not always irreducible. The primitive nth roots of unity, on the other hand, are the roots of a special irreducible polynomial (over the integers) called the nth cyclotomic polynomial, often denoted $\Phi_n(z)$ . The degree of this polynomial is precisely $\phi(n)$ , Euler's totient function, which, as we've seen, counts the number of primitive nth roots of unity.

Galois theory provides a powerful framework for understanding these cyclotomic polynomials. It demonstrates that they can be conveniently solved using radicals. While the simple expression $\sqrt[n]{1}$ might seem like a solution, it's not entirely satisfactory because it includes non-primitive roots (like 1) and doesn't neatly separate the real and imaginary parts. The solvability by radicals means that for any positive integer n, there's a specific expression built from integers using root extractions, addition, subtraction, multiplication, and division, which yields exactly the primitive nth roots of unity.

Carl Friedrich Gauss famously proved that a primitive nth root of unity can be expressed using only square roots, along with the basic arithmetic operations, if and only if a regular n-gon can be constructed using a compass and straightedge. This geometric condition is met if and only if n is a power of two or a product of a power of two and distinct Fermat primes.

Consider a primitive nth root of unity z. Its reciprocal, $1/z$ , is also a primitive nth root of unity. Now, let $r = z + 1/z$ . This value r is equal to twice the real part of z. This relationship is key to understanding $\Phi_n(z)$ , which is a reciprocal polynomial. A related polynomial, $R_n$ , whose roots are these values of r, can be derived from $\Phi_n(z)$ . The primitive nth roots of unity can then be recovered from the roots of $R_n$ by solving the quadratic equation:

$z^2 - rz + 1 = 0$

This means the real part of a primitive root is $r/2$ , and its imaginary part is $\pm i \sqrt{1 - (r/2)^2}$ . The polynomial $R_n$ is irreducible, and its degree is a power of two precisely when the regular n-gon is constructible. Otherwise, while the roots are solvable in radicals, they fall into the casus irreducibilis, meaning any radical expression will involve non-real roots.

Explicit Expressions in Low Degrees

Let's look at some specific examples:

n = 1: The cyclotomic polynomial is $\Phi_1(x) = x - 1$ . The only primitive first root of unity is 1. This root is also a non-primitive nth root for any $n > 1$ .
n = 2: $\Phi_2(x) = x + 1$ . The only primitive second root of unity is -1, which is also a non-primitive nth root for any even $n > 2$ . These two cases (n=1 and n=2) cover all real roots of unity.
n = 3: $\Phi_3(x) = x^2 + x + 1$ . The primitive third (cube) roots of unity are the roots of this quadratic polynomial: $\frac{-1 \pm i\sqrt{3}}{2}$
n = 4: $\Phi_4(x) = x^2 + 1$ . The two primitive fourth roots of unity are $i$ and $-i$ .
n = 5: $\Phi_5(x) = x^4 + x^3 + x^2 + x + 1$ . The four primitive fifth roots of unity are the roots of this quartic polynomial. They can be explicitly expressed using radicals: $\frac{\varepsilon\sqrt{5}-1}{4} \pm i\frac{\sqrt{10+2\varepsilon\sqrt{5}}}{4}$ where $\varepsilon$ can be either 1 or -1.
n = 6: $\Phi_6(x) = x^2 - x + 1$ . The two primitive sixth roots of unity are the negatives (and also the square roots) of the primitive cube roots: $\frac{1 \pm i\sqrt{3}}{2}$
n = 7: Since 7 is not a Fermat prime, the seventh roots of unity require cube roots for their expression. There are 6 primitive seventh roots, which come in complex conjugate pairs. The sum of a root and its conjugate is twice its real part. These sums are the three real roots of the cubic polynomial $r^3 + r^2 - 2r - 1$ . The primitive seventh roots themselves are then given by: $\frac{r}{2} \pm i\sqrt{1-\frac{r^2}{4}}$ where r represents the roots of the cubic. While these roots can be expressed using square and cube roots, due to the casus irreducibilis, any such expression will involve non-real cube roots.
n = 8: $\Phi_8(x) = x^4 + 1$ . The four primitive eighth roots of unity are the square roots of the primitive fourth roots ( $\pm i$ ): $\pm \frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2}$
n = 17: For the 17th root of unity, the real part is particularly notable and relates to the constructibility of the Heptadecagon.

Periodicity

Let z be a primitive nth root of unity. The sequence of its powers, $\dots, z^{-1}, z^0, z^1, \dots$ , is n-periodic. This is because $z^{j+n} = z^j z^n = z^j \cdot 1 = z^j$ for any integer j.

Furthermore, we can consider n such sequences, indexed by $k = 1, \dots, n$ . For each k, the sequence $s_k = (\dots, z^{k \cdot (-1)}, z^{k \cdot 0}, z^{k \cdot 1}, \dots)$ is also n-periodic. The set $\{s_1, \dots, s_n\}$ forms a basis for the linear space of all n-periodic sequences of complex numbers. This is a profound statement: any n-periodic sequence can be uniquely represented as a linear combination of powers of a primitive nth root of unity:

$x_j = \sum_{k} X_k \cdot z^{k \cdot j} = X_1 z^{1 \cdot j} + \dots + X_n z^{n \cdot j}$

for some complex coefficients $X_1, \dots, X_n$ .

This is essentially a form of Fourier analysis. If we consider j as a discrete time variable, then k represents a frequency, and $X_k$ represents a complex amplitude.

If we choose $z = e^{2\pi i / n} = \cos(2\pi/n) + i \sin(2\pi/n)$ as our primitive nth root, then $x_j$ can be expressed as a linear combination of sines and cosines:

$x_j = \sum_k A_k \cos \frac{2\pi jk}{n} + \sum_k B_k \sin \frac{2\pi jk}{n}$

This is precisely the form of a discrete Fourier transform.

Summation

Let SR(n) denote the sum of all the nth roots of unity, both primitive and non-primitive. Then:

$\text{SR}(n) = \begin{cases} 1 & \text{if } n=1 \\ 0 & \text{if } n>1 \end{cases}$

This result follows directly from Vieta's formulas. The nth roots of unity are the roots of the polynomial $X^n - 1$ . The sum of these roots is the coefficient of the $X^{n-1}$ term, which is 1 if $n=1$ (for $X-1$ ) and 0 if $n>1$ .

Alternatively, for $n>1$ , let $S$ be the set of nth roots of unity. Since $S$ forms a group under multiplication, for any root $z \neq 1$ , the set $zS$ is identical to $S$ . Therefore, the sum of elements in $S$ satisfies $z \cdot \text{SR}(n) = \text{SR}(n)$ , which implies $\text{SR}(n) = 0$ .

Now, let SP(n) be the sum of all the primitive nth roots of unity. It turns out that:

$\text{SP}(n) = \mu(n)$

where $\mu(n)$ is the Möbius function. This connection arises from the relationship between the sets of all roots $R(n)$ and primitive roots $P(n)$ : $R(n)$ is the disjoint union of $P(d)$ for all divisors d of n. This leads to:

$\text{SR}(n) = \sum_{d\,|\,n} \text{SP}(d)$

Applying the Möbius inversion formula to this equation yields:

$\text{SP}(n) = \sum_{d\,|\,n} \mu(d) \text{SR}\left(\frac{n}{d}\right)$

In this sum, $\text{SR}(n/d) = 0$ whenever $d < n$ (because $n/d > 1$ ). The only non-zero term occurs when $d = n$ , where $\text{SR}(n/n) = \text{SR}(1) = 1$ . Thus, $\text{SP}(n) = \mu(n) \cdot 1 = \mu(n)$ .

This result is a specific instance, $c_n(1)$ , of Ramanujan's sum $c_n(s)$ , which is defined as the sum of the sth powers of the primitive nth roots of unity:

$c_n(s) = \sum_{\substack{a=1 \\ \gcd(a,n)=1}}^{n} e^{2\pi i \frac{a}{n} s}$

Orthogonality

From the summation formulas, we can derive an important orthogonality relationship. For indices $j$ and $j'$ ranging from 1 to n:

$\sum_{k=1}^{n} \overline{z^{j \cdot k}} \cdot z^{j' \cdot k} = n \cdot \delta_{j,j'}$

Here, $\delta_{j,j'}$ is the Kronecker delta (which is 1 if $j=j'$ and 0 otherwise), and z is any primitive nth root of unity.

This property is fundamental to the discrete Fourier transform. The n × n matrix U with entries $U_{j,k} = n^{-1/2} \cdot z^{j \cdot k}$ defines this transform. While a naive computation of the transform using Gaussian elimination would take $O(n^3)$ operations, the orthogonality property reveals that U is a unitary matrix. This means its inverse is simply its complex conjugate:

$\sum_{k=1}^{n} \overline{U_{j,k}} \cdot U_{k,j'} = \delta_{j,j'}$

This insight, first noted by Gauss in the context of trigonometric interpolation, allows for efficient computation. Applying U or its inverse to a vector typically takes $O(n^2)$ operations. The development of fast Fourier transform algorithms further reduces this to a remarkable $O(n \log n)$ operations.

Cyclotomic Polynomials

Main article: Cyclotomic polynomial

The polynomial $p(z) = z^n - 1$ has the nth roots of unity as its zeros, each with multiplicity of 1. The nth cyclotomic polynomial, denoted $\Phi_n(z)$ , is defined such that its zeros are precisely the primitive nth roots of unity, each with multiplicity 1.

$\Phi_n(z) = \prod_{k=1}^{\varphi(n)} (z - z_k)$

where $z_1, z_2, \dots, z_{\varphi(n)}$ are the primitive nth roots of unity, and $\varphi(n)$ is Euler's totient function. A key property is that $\Phi_n(z)$ has integer coefficients and is an irreducible polynomial over the rational numbers. For the specific case where n is prime, this irreducibility can be demonstrated using Eisenstein's criterion applied to a related polynomial derived using the binomial theorem.

Every nth root of unity is a primitive dth root of unity for exactly one positive divisor d of n. This fundamental relationship leads to the factorization of $z^n - 1$ :

$z^n - 1 = \prod_{d\,|\,n} \Phi_d(z)$

This formula provides the factorization of $z^n - 1$ into irreducible factors. Here are a few examples:

$z^1 - 1 = z - 1$
$z^2 - 1 = (z - 1)(z + 1)$
$z^3 - 1 = (z - 1)(z^2 + z + 1)$
$z^4 - 1 = (z - 1)(z + 1)(z^2 + 1)$
$z^5 - 1 = (z - 1)(z^4 + z^3 + z^2 + z + 1)$
$z^6 - 1 = (z - 1)(z + 1)(z^2 + z + 1)(z^2 - z + 1)$
$z^7 - 1 = (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1)$
$z^8 - 1 = (z - 1)(z + 1)(z^2 + 1)(z^4 + 1)$

Applying the Möbius inversion formula to the factorization yields an explicit expression for the cyclotomic polynomial:

$\Phi_n(z) = \prod_{d\,|\,n} \left(z^{\frac{n}{d}} - 1\right)^{\mu(d)} = \prod_{d\,|\,n} \left(z^d - 1\right)^{\mu\left(\frac{n}{d}\right)}$

where $\mu$ is the Möbius function. Using this, we can derive the first few cyclotomic polynomials:

$\Phi_1(z) = z - 1$
$\Phi_2(z) = (z^2 - 1)(z - 1)^{-1} = z + 1$
$\Phi_3(z) = (z^3 - 1)(z - 1)^{-1} = z^2 + z + 1$
$\Phi_4(z) = (z^4 - 1)(z^2 - 1)^{-1} = z^2 + 1$
$\Phi_5(z) = (z^5 - 1)(z - 1)^{-1} = z^4 + z^3 + z^2 + z + 1$
$\Phi_6(z) = (z^6 - 1)(z^3 - 1)^{-1}(z^2 - 1)^{-1} = z^2 - z + 1$
$\Phi_7(z) = (z^7 - 1)(z - 1)^{-1} = z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$
$\Phi_8(z) = (z^8 - 1)(z^4 - 1)^{-1} = z^4 + 1$

For a prime number p, all the pth roots of unity except 1 are primitive. Therefore:

$\Phi_p(z) = \frac{z^p - 1}{z - 1} = \sum_{k=0}^{p-1} z^k$

Substituting any integer $z \ge 2$ into this sum results in a repunit in base z. This implies that a necessary (though not sufficient) condition for a repunit to be prime is that its length must be prime.

It's worth noting that not all cyclotomic polynomials have coefficients restricted to 0, 1, or -1. The first instance where this occurs is $\Phi_{105}(z)$ . The complexity of the coefficients is related to the number of distinct odd prime factors of n. If n has one or two odd prime factors, the coefficients are indeed limited to 0, 1, or -1. The number 105, being the product of the three smallest odd primes (3, 5, and 7), is the first candidate for a cyclotomic polynomial with coefficients beyond $\pm 1$ . A theorem by Schur states that cyclotomic polynomials can have arbitrarily large coefficients in absolute value. Specifically, if $n = p_1 p_2 \dots p_t$ where $p_1 < p_2 < \dots < p_t$ are odd primes, $p_1 + p_2 > p_t$ , and t is odd, then $1-t$ appears as a coefficient in $\Phi_n(z)$ .

There are also known restrictions on the values that cyclotomic polynomials can take when evaluated at integer points. For instance, if p is prime, then $d \mid \Phi_p(d)$ if and only if $d \equiv 1 \pmod p$ .

Cyclotomic polynomials are inherently solvable in radicals, as roots of unity themselves are radicals. In fact, more refined radical expressions exist where every possible choice of radical values yields a primitive nth root of unity. Gauss demonstrated this in 1797, and efficient algorithms exist for computing these expressions.

Cyclic Groups

The nth roots of unity, under multiplication, form a cyclic group of order n. These groups represent all possible finite subgroups of the multiplicative group of complex numbers. A primitive nth root of unity serves as a generator for this cyclic group.

Furthermore, the nth roots of unity provide an irreducible representation for any cyclic group of order n. The orthogonality relationship previously discussed can also be derived from group-theoretic principles related to character groups.

The entries of the eigenvectors of any circulant matrix (matrices invariant under cyclic shifts) are precisely the roots of unity. This is another consequence of group representation theory, related to Bloch's theorem. page needed In the specific case of a circulant Hermitian matrix, such as a discretized one-dimensional Laplacian with periodic boundary conditions page needed, the orthogonality of eigenvectors of Hermitian matrices naturally leads to the orthogonality of roots of unity.

Cyclotomic Fields

Main article: Cyclotomic field

By adjoining a primitive nth root of unity to the field of rational numbers $\mathbb{Q}$ , we obtain the nth cyclotomic field:

$\mathbb{Q}(\exp(2\pi i/n))$

This field encompasses all nth roots of unity and is the splitting field of the nth cyclotomic polynomial over $\mathbb{Q}$ . The field extension $\mathbb{Q}(\exp(2\pi i/n)) / \mathbb{Q}$ has a degree of $\phi(n)$ , and its Galois group is naturally isomorphic to the multiplicative group of units of the ring $\mathbb{Z}/n\mathbb{Z}$ .

Since the Galois group of $\mathbb{Q}(\exp(2\pi i/n)) / \mathbb{Q}$ is abelian, this constitutes an abelian extension. In fact, every subfield of a cyclotomic field is an abelian extension of the rationals. Consequently, every nth root of unity can be expressed using k-th roots, where k does not exceed $\phi(n)$ . Galois theory allows for explicit descriptions of these extensions using Gaussian periods. This theory, pioneered by Gauss in his Disquisitiones Arithmeticae, predates Galois's work. page needed

Conversely, every abelian extension of the rationals is a subfield of some cyclotomic field. This is the essence of a theorem by Leopold Kronecker, often referred to as the Kronecker–Weber theorem, with completion by Weber.

Relation to Quadratic Integers

In the complex plane, the points representing the fifth roots of unity (red) and the sums of a fifth root and its complex conjugate (black) illustrate interesting relationships. The corners of the squares represent the eighth roots of unity.

For $n=1$ and $n=2$ , the roots of unity (1 and -1) are ordinary integers.
For $n=3$ and $n=6$ , the roots of unity are Eisenstein integers, characterized by the discriminant $D=-3$ .
For $n=4$ , the roots are Gaussian integers, with $D=-1$ . These include the imaginary unit, i.

For several other values of n, while the primitive roots of unity themselves might not be quadratic integers, the sum of a root z and its complex conjugate $\bar{z}$ (which is also an nth root of unity) is a quadratic integer.

For $n=5$ and $n=10$ , none of the non-real roots (which satisfy a quartic equation) are quadratic integers. However, the sum $z + \bar{z} = 2 \cdot \text{Re}(z)$ belongs to the ring $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$ (with $D=5$ ). For two pairs of non-real fifth roots, these sums are the inverse of the golden ratio ( $\phi$ ) and its negative.
For $n=8$ , the sum $z + \bar{z}$ can take values of 0, $\pm 2$ , or $\pm \sqrt{2}$ (with $D=2$ ).
For $n=12$ , the sum $z + \bar{z}$ can be 0, $\pm 1$ , $\pm 2$ , or $\pm \sqrt{3}$ (with $D=3$ ).

Notes

[1] Hadlock, Charles R. (2000). Field Theory and Its Classical Problems, Volume 14. Cambridge University Press. pp. 84–86. ISBN 978-0-88385-032-9.
[2] Lang, Serge (2002). "Roots of unity". Algebra. Springer. pp. 276–277. ISBN 978-0-387-95385-4.
[3] Meserve, Bruce E. (1982). Fundamental Concepts of Algebra. Dover Publications. p. 52.
[4] Moskowitz, Martin A. (2003). Adventure in Mathematics. World Scientific. p. 36. ISBN 9789812794949.
[5] Lidl, Rudolf; Pilz, Günter (1984). Applied Abstract Algebra. Undergraduate Texts in Mathematics. Springer. p. 149. doi:10.1007/978-1-4615-6465-2. ISBN 978-0-387-96166-8.
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[11] Lehmer, Emma (1936). "On the magnitude of the coefficients of the cyclotomic polynomial". Bulletin of the American Mathematical Society. 42 (6): 389–392. doi:10.1090/S0002-9904-1936-06309-3.
[12] Landau, Susan; Miller, Gary L. (1985). "Solvability by radicals is in polynomial time". Journal of Computer and System Sciences. 30 (2): 179–208. doi:10.1016/0022-0000(85)90013-3.
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[17] The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but his work wasn't published until 1846.