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Root of unity

In mathematics, particularly in complex analysis and abstract algebra, the nth roots of unity are the complex numbers \zeta satisfying the equation \zeta^n = 1, where n is a positive integer; these are explicitly given by \zeta_k = e^{2\pi i k / n} = \cos(2\pi k / n) + i \sin(2\pi k / n) for k = 0, 1, \dots, n-1.^[1] These points lie equally spaced on the unit circle in the complex plane, forming the vertices of a regular n-gon inscribed in that circle.^[2] Under complex multiplication, they constitute a cyclic group of order n, with 1 as the identity element and the sum of all nth roots equaling zero when n > 1.^[1] A primitive nth root of unity is any \zeta_k where \gcd(k, n) = 1, generating the entire group through its powers; the number of such primitive roots is given by Euler's totient function \phi(n).^[1] The minimal polynomial for the primitive nth roots over the rationals is the nth cyclotomic polynomial \Phi_n(x), which is monic, irreducible, and has integer coefficients, playing a central role in the study of cyclotomic fields.^[1] In field theory, adjoining a primitive nth root to a base field F (of characteristic not dividing n) yields a Galois extension whose Galois group is isomorphic to the multiplicative group of units modulo n, (\mathbb{Z}/n\mathbb{Z})^\times, highlighting their importance in understanding algebraic number fields and solvability by radicals.^[3] Roots of unity have broad applications across mathematics and related fields: in geometry, they parameterize the symmetries of regular polygons; in trigonometry, they express cosine and sine values via relations like \cos(2\pi / n) = (\zeta + \zeta^{-1})/2; and in signal processing, the nth roots serve as basis functions for the discrete Fourier transform, enabling efficient analysis of periodic signals in audio and imaging.^[1]^[4] They also appear in the roots-of-unity filter for evaluating sums and in the representation theory of quantum groups at roots of unity, connecting to modular forms and physics.^[5]

Definition and Fundamentals

Definition

A root of unity is a complex number \zeta such that \zeta^n = 1 for some positive integer n. The nth roots of unity are precisely the solutions to the equation z^n - 1 = 0 in the complex numbers.^[6] A primitive nth root of unity is an nth root of unity of exact order n, meaning \zeta^k \neq 1 for all positive integers k < n. The standard primitive nth root of unity is denoted \omega_n = e^{2\pi i / n}.^[6] The concept of roots of unity was systematically introduced by Carl Friedrich Gauss in his 1801 work Disquisitiones Arithmeticae, particularly in the context of cyclotomic fields.^[7] For n=1, the only root of unity is $1. For n=2, the roots are $1 and -1. For n=3, the roots are the three solutions to z^3 = 1.^[6]

Elementary Properties

All roots of unity lie on the unit circle in the complex plane, as any such z satisfies |z|^n = |1| = 1, implying |z| = 1.^[8] The nth roots of unity are equally spaced around this circle, forming the vertices of a regular n-gon, with angular positions $2\pi k / n for k = 0, 1, \dots, n-1.^[8] Considering the polynomial equation z^n - 1 = 0, Vieta's formulas yield key symmetric properties of the roots. For n > 1, the sum of all nth roots of unity is zero, corresponding to the zero coefficient of z^{n-1}.^[9] The product of all nth roots of unity is (-1)^{n+1}, derived from the constant term -1 with the sign alternation (-1)^n.^[9] Every root of unity is an algebraic integer, as it satisfies a monic polynomial equation with integer coefficients—specifically, its minimal polynomial over the rationals divides the nth cyclotomic polynomial, which is monic and has integer coefficients.^[10] If [\zeta](/page/Zeta) is a primitive nth root of unity, then the power \zeta^m is a primitive dth root of unity, where d = n / \gcd(m, n).^[11]

Geometric and Trigonometric Aspects

Trigonometric Representation

The nth roots of unity can be expressed in trigonometric form using Euler's formula, which states that e^{i\theta} = \cos \theta + i \sin \theta for real \theta.^[12] The solutions to z^n = 1 are thus given by z_k = e^{2\pi i k / n} = \cos(2\pi k / n) + i \sin(2\pi k / n) for integers k = 0, 1, \dots, n-1.^[12] This representation places each root on the unit circle in the complex plane, with the real part \cos(2\pi k / n) and imaginary part \sin(2\pi k / n) serving as the Cartesian coordinates.^[13] A primitive nth root of unity, denoted \omega_n, is the root with the smallest positive argument, given by \omega_n = \cos(2\pi / n) + i \sin(2\pi / n).^[12] All other nth roots are powers of this primitive root: z_k = \omega_n^k. Geometrically, the nth roots of unity correspond to the vertices of a regular n-gon inscribed in the unit circle centered at the origin, equally spaced at angular intervals of $2\pi / n. De Moivre's theorem, which states that [\cos \theta + i \sin \theta]^m = \cos(m\theta) + i \sin(m\theta) for integer m, applies directly to powers of roots of unity.^[14] Raising a root z_k to the mth power rotates it by multiples of $2\pi m / n around the unit circle, preserving the cyclic structure.^[15] This trigonometric form underscores the rotational symmetry inherent in roots of unity.^[16]

Periodicity

The order of a root of unity \zeta is defined as the smallest positive integer m such that \zeta^m = 1.^[17] For an nth root of unity, this order divides n.^[18] In particular, for a primitive nth root of unity (order n), \zeta^k = 1 precisely when n divides k.^[17] The powers of an nth root of unity \zeta exhibit periodicity with period n, as \zeta^{k + n} = \zeta^k \cdot \zeta^n = \zeta^k \cdot 1 = \zeta^k for any integer k.^[17] Consequently, the sequence of powers \zeta^0, \zeta^1, \dots, \zeta^{n-1} repeats indefinitely every n steps. This repetition in the exponents directly corresponds to arithmetic modulo n, where \zeta^k = \zeta^{k \mod n}, mirroring the cyclic structure of the ring \mathbb{Z}/n\mathbb{Z}.^[17] While each root of unity has a finite order dividing some n, the collection of all roots of unity forms an infinite group under multiplication, with no universal period encompassing every element.^[19] Instead, individual roots are torsion elements of finite order. For instance, the cube roots of unity satisfy \omega^3 = 1, so their powers cycle every 3 steps: \omega^0 = 1, \omega^1 = \omega, \omega^2 = \omega^2, \omega^3 = 1, and so on.^[17]

Algebraic Representations

Explicit Formulas for Low Degrees

For the first root of unity, corresponding to n=1, the only solution to z^1 = 1 is z = 1.^[20] For n=2, the equation z^2 = 1 factors as (z-1)(z+1) = 0, yielding the roots z = \pm 1.^[20] The third roots of unity satisfy z^3 = 1, or (z-1)(z^2 + z + 1) = 0. The quadratic factor z^2 + z + 1 = 0 has discriminant $1 - 4 = -3, so the primitive roots are z = \frac{-1 \pm \sqrt{-3}}{2} = -\frac{1}{2} \pm i \frac{\sqrt{3}}{2}.^[20] For n=4, the equation z^4 = 1 factors as (z^2 - 1)(z^2 + 1) = 0, giving roots z = \pm 1 and z = \pm \sqrt{-1} = \pm i.^[20] The fifth roots of unity solve z^5 = 1, or (z-1)\Phi_5(z) = 0 where \Phi_5(z) = z^4 + z^3 + z^2 + z + 1 = 0. These primitive roots can be expressed using square roots involving the golden ratio \phi = \frac{1 + \sqrt{5}}{2}. Specifically, one primitive root is \cos\frac{2\pi}{5} + i \sin\frac{2\pi}{5} = \frac{\sqrt{5} - 1}{4} + i \frac{\sqrt{10 + 2\sqrt{5}}}{4}, and another is \cos\frac{4\pi}{5} + i \sin\frac{4\pi}{5} = \frac{-\sqrt{5} - 1}{4} + i \frac{\sqrt{10 - 2\sqrt{5}}}{4}; the remaining roots are their complex conjugates and powers.^[21]^[20] These cases for n \leq 5 are solvable using only square roots (including of negative numbers), yielding compact radical expressions. In contrast, for larger n such as n=7, while still expressible by radicals due to the abelian Galois group of cyclotomic extensions, the formulas require higher-degree roots like cube roots and more nested iterations.^[20]

General Algebraic Expressions

The nth roots of unity are the complex numbers satisfying the equation z^n - 1 = 0, which factors uniquely over the rationals as z^n - 1 = \prod_{d \mid n} \Phi_d(z), where each \Phi_d(z) is the dth cyclotomic polynomial.^[6] The primitive nth roots of unity are precisely the roots of the irreducible cyclotomic polynomial \Phi_n(z), which has degree \phi(n) and is monic with integer coefficients.^[6] For general n, explicit algebraic expressions for the roots require solving these irreducible polynomials over \mathbb{Q}, but no simple closed-form formula exists using only finitely many arithmetic operations and root extractions that applies uniformly across all n.^[22] Although the Galois group of \Phi_n(z) over \mathbb{Q} is abelian and thus solvable—implying the roots are expressible by radicals—these expressions become increasingly complex for larger n, involving nested radicals of degrees matching the structure of (\mathbb{Z}/n\mathbb{Z})^\times.^[6] The Abel-Ruffini theorem establishes that no general solution by radicals exists for arbitrary polynomials of degree at least 5, and while cyclotomic polynomials evade this for specific cases due to their solvable Galois groups, the lack of a uniform radical formula for \Phi_n(z) when \phi(n) \geq 5 (as occurs for n ≥ 7) underscores the challenge in obtaining elementary expressions.^[23] Attempts to express roots via nested radicals highlight these limitations; for instance, the primitive 7th roots of unity satisfy a degree-6 irreducible polynomial and can be written using nested cube roots and square roots, but the resulting formula is highly intricate and non-elementary in structure.^[22] For broader n, such constructions rely on Lagrange resolvents or computational methods to denest radicals, yet they do not yield a general pattern beyond low degrees.^[24] In practice, numerical approximations often supplement algebraic efforts, particularly for the real parts of the roots. The cosine values \cos(2\pi k / n) for k = 1, ..., \lfloor n/2 \rfloor can be approximated via the Taylor series expansion \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots, where x = 2π/n, providing high precision for large n without solving the full polynomial. This series converges rapidly for small x, establishing key quantitative context for the distribution of roots on the unit circle when exact algebraic forms are infeasible. Overall, expressing general nth roots of unity algebraically ties directly to resolving the minimal polynomials \Phi_n(z) over \mathbb{Q}, where irreducibility ensures the roots generate a degree-\phi(n) extension, but practical computation favors hybrid algebraic-numeric approaches for n beyond small values.^[6]

Group-Theoretic Structure

nth Roots of Unity as a Group

The set of all nth roots of unity, denoted \mu_n = \{\zeta \in \mathbb{C} \mid \zeta^n = 1\}, forms a finite abelian group under complex multiplication, specifically a cyclic group of order n.^[17] This structure arises because the roots are the solutions to the equation z^n - 1 = 0, and their multiplication closes within the set, with the identity element being $1(corresponding to\zeta^0$).^[25] A generator of this group is any primitive nth root of unity \omega_n, such as \omega_n = e^{2\pi i / n}, which has multiplicative order exactly n. The elements of \mu_n are then precisely the powers \{\omega_n^k \mid k = 0, 1, \dots, n-1\}, confirming the cyclic nature generated by a single element.^[17]^[26] The group \mu_n is isomorphic to the additive group \mathbb{Z}/n\mathbb{Z} of integers modulo n, via the map k \mapsto \omega_n^k for k \in \{0, 1, \dots, n-1\}. This isomorphism preserves the group operation: multiplication in \mu_n corresponds to addition modulo n in \mathbb{Z}/n\mathbb{Z}.^[25]^[17] For each positive divisor d of n, the set of dth roots of unity \mu_d forms a subgroup of \mu_n of order d and index n/d. These subgroups are unique for each d, as they consist of the elements in \mu_n whose order divides d.^[17]^[26] By Lagrange's theorem applied to the finite group \mu_n, the order of every element divides n, meaning that if \zeta \in \mu_n has order m, then m \mid n. This implies that all proper subgroups of \mu_n correspond exactly to the divisors of n, reinforcing the cyclic structure.^[17]

All Roots of Unity as a Group

The set of all roots of unity in the complex numbers, often denoted by \mu or \mu_\infty, is the union over all positive integers n of the sets of nth roots of unity. This set forms a subgroup of the multiplicative group \mathbb{C}^\times of nonzero complex numbers, as the product of two roots of unity (of orders m and n) is a root of unity of order dividing \mathrm{lcm}(m,n), and the inverse of a root of unity of order n is its complex conjugate, which is also a root of unity of order dividing n.^[27] Moreover, \mu is precisely the torsion subgroup of \mathbb{C}^\times, consisting of all elements of finite order, since any z \in \mathbb{C}^\times satisfying z^k = 1 for some positive integer k must satisfy |z| = 1 and thus lie on the unit circle.^[27] As a subgroup of the unit circle group S^1 = \{ z \in \mathbb{C} : |z| = 1 \}, \mu is countable, being the countable union of the finite sets of nth roots of unity for each n. Despite its countability, \mu is dense in S^1: for any w \in S^1 and \epsilon > 0, there exist integers m, n such that \left| \exp(2\pi i m / n) - w \right| < \epsilon, since the rational multiples of $2\pi are dense in [0, 2\pi) by the density of \mathbb{Q} in \mathbb{R}.^[28] Every element of \mu has finite order, and for any positive integer n, \mu contains the full cyclic subgroup of nth roots of unity. Algebraically, \mu can be viewed as the direct limit of the directed system of finite cyclic groups \mu_n \cong \mathbb{Z}/n\mathbb{Z} (under multiplication), where the maps \mu_m \to \mu_n exist whenever m divides n via the natural inclusion.^[29] As abstract groups, \mu is isomorphic to the additive group \mathbb{Q}/\mathbb{Z} via the exponential map \mapsto \exp(2\pi i q) for q \in \mathbb{Q}/\mathbb{Z}, which is a group homomorphism sending torsion elements to roots of unity and is bijective since every root of unity is \exp(2\pi i r) for some rational r.^[27] Furthermore, \mathbb{Q}/\mathbb{Z} \cong \bigoplus_p \mathbb{Z}(p^\infty), where the direct sum is over all primes p and \mathbb{Z}(p^\infty) is the Prüfer p-group, the inductive limit of the cyclic groups \mathbb{Z}/p^k \mathbb{Z} (equivalently, multiplicatively, the group of all p^kth roots of unity for k \geq 0).^[30] This decomposition reflects the primary decomposition of the torsion in \mathbb{Q}/\mathbb{Z}, with each Prüfer component capturing the p-primary torsion elements.

Primitive Roots and Their Role

A primitive nth root of unity is defined as an nth root of unity \zeta whose multiplicative order is exactly n, meaning \zeta^n = 1 but \zeta^k \neq 1 for any positive integer k < n.^[17] Equivalently, \zeta is primitive if \zeta^k = 1 implies that n divides k.^[17] This characterization ensures that the minimal period of \zeta matches n, distinguishing it from roots of lower order.^[31] The number of primitive nth roots of unity equals \phi(n), where \phi denotes Euler's totient function, which counts the integers from 1 to n that are coprime to n.^[17] If \zeta is any primitive nth root, the full set of primitive nth roots consists of \{\zeta^k : 1 \leq k \leq n, \gcd(k, n) = 1\}.^[17] This count arises from the structure of the cyclic group of nth roots and can be derived via Möbius inversion applied to the relation n = \sum_{d \mid n} \phi(d), yielding the explicit formula

\phi(n) = \sum_{d \mid n} \mu(d) \frac{n}{d},

where \mu is the Möbius function, defined as \mu(m) = 0 if m has a squared prime factor, \mu(m) = 1 if m has an even number of distinct prime factors, and \mu(m) = -1 if odd.^[32] Primitive nth roots play a central role as generators of the multiplicative group of all nth roots of unity, which is cyclic of order n. Specifically, for any primitive \zeta, the powers \zeta^0, \zeta^1, \dots, \zeta^{n-1} exhaustively produce every nth root of unity.^[17] For example, when n = p is prime, \phi(p) = p-1, so there are p-1 primitive pth roots, each generating the full group.^[31]

Cyclotomic Theory

Cyclotomic Polynomials

The nth cyclotomic polynomial, denoted \Phi_n(z), is defined as the monic polynomial whose roots are precisely the primitive nth roots of unity, that is,

\Phi_n(z) = \prod (z - \zeta),

where the product runs over all primitive nth roots of unity \zeta.^[10]^[33] The degree of \Phi_n(z) is given by Euler's totient function \phi(n), which counts the number of integers up to n that are coprime to n. This follows directly from the fact that there are exactly \phi(n) primitive nth roots of unity.^[10]^[33] A fundamental property is the factorization of the nth cyclotomic polynomial in relation to the polynomial z^n - 1:

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

where the product is over all positive divisors d of n. This decomposition arises because the roots of z^n - 1 are all nth roots of unity, partitioned according to their orders.^[10]^[33] From this factorization, a recursive formula for \Phi_n(z) can be derived:

\Phi_n(z) = \frac{z^n - 1}{\prod_{\substack{d \mid n \\ d < n}} \Phi_d(z)}.

This allows computation of \Phi_n(z) using previously computed cyclotomic polynomials for proper divisors of n.^[34]^[33] The cyclotomic polynomials \Phi_n(z) are irreducible over the rationals \mathbb{Q}. For prime p, this was first proved by Gauss in 1801 using properties of roots and symmetric functions to show that any factorization would contradict the minimal polynomial degree. In general, irreducibility follows from criteria such as Eisenstein's (applied after the substitution z \mapsto z + 1 for prime powers) or more advanced methods involving substitutions and field extensions, as established by later mathematicians including Dedekind and others.^[34] Explicit examples for small n illustrate these properties. For n=1, \Phi_1(z) = z - 1. For n=2, \Phi_2(z) = z + 1. For n=3, \Phi_3(z) = z^2 + z + 1. For n=4, \Phi_4(z) = z^2 + 1. Each has integer coefficients and is monic of degree \phi(n).

Cyclotomic Fields

The cyclotomic field \mathbb{Q}(\zeta_n) is the extension of the rational numbers \mathbb{Q} obtained by adjoining a primitive nth root of unity \zeta_n, and it can be explicitly constructed as the quotient ring \mathbb{Q} / (\Phi_n(x)), where \Phi_n(x) is the nth cyclotomic polynomial.^[6] This field has degree \phi(n) over \mathbb{Q}, where \phi denotes Euler's totient function, reflecting the minimal polynomial degree of \zeta_n over \mathbb{Q}.^[6] A power basis for \mathbb{Q}(\zeta_n) as a vector space over \mathbb{Q} is given by \{1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{\phi(n)-1}\}.^[6] The extension \mathbb{Q}(\zeta_n)/\mathbb{Q} is Galois, meaning it is both normal and separable, with the Galois group \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) isomorphic to the multiplicative group of units modulo n, denoted (\mathbb{Z}/n\mathbb{Z})^\times.^[6] This isomorphism arises from the action of automorphisms on \zeta_n, sending it to \zeta_n^k for k coprime to n.^[6] The abelian nature of this Galois group underscores the simplicity of cyclotomic extensions compared to more general number fields.^[35] For any positive divisor d of n, the cyclotomic field \mathbb{Q}(\zeta_d) is a subfield of \mathbb{Q}(\zeta_n).^[6] This inclusion follows from the fact that \zeta_n^{n/d} is a primitive dth root of unity, ensuring that all lower-order cyclotomic fields embed naturally within higher ones.^[6] In the ring of integers of \mathbb{Q}(\zeta_n), ramification occurs only at the finite primes dividing n; all other primes remain unramified.^[36] This property highlights the localized arithmetic complexity of cyclotomic fields, with unramified primes splitting according to the Frobenius elements in the Galois group.^[36] Historically, cyclotomic fields played a pivotal role in Carl Friedrich Gauss's 1796 proof of the constructibility of the regular 17-gon using compass and straightedge, achieved by explicitly solving for the real subfield of \mathbb{Q}(\zeta_{17}) through quadratic extensions.^[37] This breakthrough demonstrated that a regular heptadecagon could be constructed, leveraging the degree \phi(17) = 16 = 2^4 to reduce the problem to successive square roots.^[37]

Analytic and Summation Properties

Summation Formulas

One fundamental summation formula involving roots of unity arises from the geometric series. For a primitive nth root of unity \zeta = e^{2\pi i / n} with n > 1, the sum \sum_{k=0}^{n-1} \zeta^k = 0. This follows from the formula for the sum of a finite geometric series: \sum_{k=0}^{n-1} r^k = (1 - r^n)/(1 - r) for r \neq 1, where r = \zeta and \zeta^n = 1, yielding (1 - 1)/(1 - \zeta) = 0.^[1] More generally, consider power sums over all nth roots of unity. Let \omega = e^{2\pi i / n} be a primitive nth root of unity. The sum p_m = \sum_{k=0}^{n-1} \omega^{k m} equals n if n divides m, and $0otherwise. This result holds because\omega^mis a primitivedth root of unity where d = n / \gcd(n, m); the [geometric series](/page/Geometric_series) sum is then nonly when\omega^m = 1(i.e.,n \mid m

), and &#36;0

otherwise.^[1] A related summation is the Ramanujan sum, which involves only the primitive nth roots of unity. Defined as c_n(m) = \sum_{\substack{k=1 \\ \gcd(k,n)=1}}^n e^{2\pi i k m / n}, this equals the sum of the mth powers of the primitive nth roots of unity. Its closed form is c_n(m) = \mu(n / d) \cdot \phi(n) / \phi(n / d), where d = \gcd(m, n), \mu is the Möbius function, and \phi is Euler's totient function.^[38] These formulas find applications in discrete Fourier analysis, where the power sum orthogonality underpins the inversion of the discrete Fourier transform via sums over roots of unity.^[1]

Orthogonality Relations

The nth roots of unity form a set of characters for the cyclic group \mathbb{Z}/n\mathbb{Z}, which are group homomorphisms from \mathbb{Z}/n\mathbb{Z} to the multiplicative group \mathbb{C}^* of nonzero complex numbers. Specifically, for a primitive nth root of unity \omega = e^{2\pi i / n}, the characters are given by \chi_j(k) = \omega^{j k} for j, k = 0, 1, \dots, n-1, where addition is modulo n.^[39] These characters satisfy orthogonality relations with respect to the inner product on functions from \mathbb{Z}/n\mathbb{Z} to \mathbb{C}, defined as \langle f, g \rangle = \sum_{k=0}^{n-1} f(k) \overline{g(k)}. In particular, the inner product between distinct characters is zero: \sum_{k=0}^{n-1} \omega^{k(j - l)} = n \delta_{j l \mod n}, where \delta_{j l \mod n} is the Kronecker delta, equal to 1 if j \equiv l \pmod{n} and 0 otherwise.^[40] This relation follows from the geometric series sum when j \not\equiv l \pmod{n} and the trivial sum when j \equiv l \pmod{n}.^[39] The set of characters \{\chi_j \mid j = 0, 1, \dots, n-1\} forms a complete orthogonal basis for the vector space of all functions from \mathbb{Z}/n\mathbb{Z} to \mathbb{C}, which has dimension n. Any function f: \mathbb{Z}/n\mathbb{Z} \to \mathbb{C} can thus be uniquely expanded as f(k) = \sum_{j=0}^{n-1} \hat{f}(j) \chi_j(k), where the coefficients \hat{f}(j) are determined by the orthogonality. This completeness ensures that the characters diagonalize circulant matrices and convolution operators on \mathbb{Z}/n\mathbb{Z}.^[40] A key application of these orthogonality relations is the discrete Fourier transform (DFT), which decomposes a sequence a = (a_0, a_1, \dots, a_{n-1}) into its frequency components using the characters. The DFT is defined as \hat{a}_j = \frac{1}{n} \sum_{k=0}^{n-1} a_k \omega^{-k j} for j = 0, 1, \dots, n-1, and the inversion formula recovers the original sequence via a_k = \sum_{j=0}^{n-1} \hat{a}_j \omega^{k j}. This transform leverages the orthogonality to ensure invertibility and Parseval's identity, \sum_{k=0}^{n-1} |a_k|^2 = n \sum_{j=0}^{n-1} |\hat{a}_j|^2. In practice, the DFT is used for filtering periodic signals by transforming to the frequency domain, applying modifications (such as zeroing certain frequencies), and inverting the transform.^[4] The DFT also facilitates solving linear difference equations with periodic coefficients or boundary conditions on finite domains, such as those arising in numerical simulations of periodic phenomena. By transforming the equation into the frequency domain, the orthogonality decouples the variables, allowing componentwise solutions before inversion.^[41]

Advanced Algebraic Connections

Galois Groups of Primitive Roots

The Galois group of the cyclotomic extension \mathbb{Q}(\zeta_n)/\mathbb{Q}, where \zeta_n is a primitive nth root of unity, is isomorphic to the multiplicative group of units modulo n, denoted (\mathbb{Z}/n\mathbb{Z})^\times. This isomorphism arises from the fact that the extension is Galois of degree \phi(n), where \phi is Euler's totient function, and the automorphisms are determined by their action on \zeta_n. Specifically, each \sigma_a \in \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) corresponds to an integer a coprime to n via \sigma_a(\zeta_n) = \zeta_n^a, providing a faithful representation of (\mathbb{Z}/n\mathbb{Z})^\times.^[6]^[42] This group action permutes the primitive nth roots of unity by exponentiation: the set of primitive nth roots consists of \zeta_n^k for k coprime to n, and \sigma_a maps \zeta_n^k to \zeta_n^{ak}, which is again primitive since \gcd(ak, n) = 1. The action is transitive on this set, reflecting the irreducibility of the nth cyclotomic polynomial over \mathbb{Q}.^[6]^[42] The fixed fields of subgroups of (\mathbb{Z}/n\mathbb{Z})^\times correspond to subextensions of \mathbb{Q}(\zeta_n)/\mathbb{Q}, yielding intermediate cyclotomic fields \mathbb{Q}(\zeta_d) for divisors d of n. Each such subgroup H fixes the subfield generated by roots of unity of order dividing the conductor associated to H.^[6] When n = p is prime, the Galois group \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}) is cyclic of order p-1, isomorphic to (\mathbb{Z}/p\mathbb{Z})^\times, which is generated by a single element corresponding to a primitive root modulo p. This cyclic structure simplifies computations of subfields and ramification.^[42]^[6] In the broader context of class field theory, the cyclotomic extension \mathbb{Q}(\zeta_n)/\mathbb{Q} realizes the ray class field of \mathbb{Q} modulo the conductor n, with the Galois group isomorphic to the ray class group modulo n; the class number of \mathbb{Q}(\zeta_n) influences the structure of its unit group via relations like Dirichlet's class number formula, providing an entry point to more advanced abelian extensions.^[43]

Real Parts and Quadratic Integers

The maximal real subfield of the nth cyclotomic field \mathbb{Q}(\zeta_n), where \zeta_n = e^{2\pi i / n} is a primitive nth root of unity, is the subfield fixed by complex conjugation. This subfield is generated by \zeta_n + \zeta_n^{-1} = 2\cos(2\pi / n) and equals \mathbb{Q}(\cos(2\pi / n)). For n > 2, the degree of this extension over \mathbb{Q} is \phi(n)/2, where \phi is Euler's totient function.^[44] The Galois group of \mathbb{Q}(\cos(2\pi / n))/\mathbb{Q} is isomorphic to (\mathbb{Z}/n\mathbb{Z})^\times / \{\pm 1\}, the quotient of the unit group modulo n by the subgroup generated by -1, which corresponds to the action of complex conjugation in the full Galois group of \mathbb{Q}(\zeta_n)/\mathbb{Q}.^[6] When \phi(n)/2 = 2, or equivalently \phi(n) = 4, the real subfield is a real quadratic extension of \mathbb{Q}, and $2\cos(2\pi / n) is a quadratic integer generating the ring of integers in certain cases. The positive integers n satisfying \phi(n) = 4 are n=5, 8, 10, 12. For n=5, $2\cos(2\pi / 5) = (\sqrt{5} - 1)/2 satisfies the minimal polynomial x^2 + x - 1 = 0 over \mathbb{Z}, generating \mathbb{Q}(\sqrt{5}). For n=8, $2\cos(2\pi / 8) = \sqrt{2} satisfies x^2 - 2 = 0, generating \mathbb{Q}(\sqrt{2}). For n=10, $2\cos(2\pi / 10) = (\sqrt{5} + 1)/2 satisfies x^2 - x - 1 = 0, again generating \mathbb{Q}(\sqrt{5}). For n=12, $2\cos(2\pi / 12) = \sqrt{3} satisfies x^2 - 3 = 0, generating \mathbb{Q}(\sqrt{3}).^[45] The quadratic cyclotomic fields occur precisely for n=3,4,6, where [\mathbb{Q}(\zeta_n) : \mathbb{Q}] = 2. Here, \mathbb{Q}(\zeta_3) = \mathbb{Q}(\sqrt{-3}), \mathbb{Q}(\zeta_4) = \mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}), and \mathbb{Q}(\zeta_6) = \mathbb{Q}(\sqrt{-3}). In each case, the real parts \cos(2\pi k / n) for k=1,\dots,n-1 are rational (specifically, -1/2, $0, or 1/2), so they generate \mathbb{Q}

as a field. However, the primitive roots of unity themselves are quadratic integers: the &#36;3

rd and $6th roots lie in the Eisenstein integers \mathbb{Z}[\omega]with\omega = (-1 + \sqrt{-3})/2

, while the &#36;4

th roots lie in the Gaussian integers \mathbb{Z}. These roots of unity exhaust the torsion units in their respective rings of integers, forming the full unit group up to sign: \{ \pm 1, \pm \zeta_3, \pm \zeta_3^2 \} for \mathbb{Z}[\omega] and \{ \pm 1, \pm i \} for \mathbb{Z}. No other quadratic fields contain roots of unity of order greater than $2beyond\pm 1$.^[17] The element $2\cos(2\pi / n) also connects to the structure of units in real quadratic fields through multiple-angle formulas for cosine, which yield recurrence relations satisfied by powers of these elements. These relations mirror the linear recurrences arising in solutions to Pell equations x^2 - d y^2 = \pm 1 or \pm 4 in fields like \mathbb{Q}(\sqrt{d}), where d=2,3,5 as above; for instance, Chebyshev polynomials of the first kind T_m(2\cos \theta) = 2\cos(m \theta) express such units explicitly.^[46]

References

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None
### Summary of Roots of Unity Content
[2]
[PDF] roots on a circle - keith conrad
Introduction. The nth roots of unity obviously all lie on the unit circle (see Figure 1 with n = 7). Algebraic integers that are not roots of unity can also ...
[3]
[PDF] MATH 421 Lecture notes Roots of unity with special emphasis on ...
Definition: Let n ∈ N and α ∈ F. We say α is a primitive nth root of unity if α generates all of the distinct roots of the polynomial xn – 1.
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Roots of Unity - Center for Computer Research in Music and Acoustics
th roots of unity under ordinary complex multiplication. We will learn later that the $ N$ th roots of unity are used to generate all the sinusoids used by the ...
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[PDF] On The Roots of Unity in Several Complex Neutrosophic Rings
Apr 11, 2023 · Roots of unity play a basic role in the theory of algebraic extensions of fields and rings. The aim of this paper is to obtain an algorithm ...
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[PDF] cyclotomic extensions - keith conrad
For a field K, an extension of the form K(ζ), where ζ is a root of unity, is called a cyclotomic extension of K. The term cyclotomic means “circle-dividing,” ...
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Math Primer: Roots of Unity | Unknown Quantity: A Real and ...
Carl Friedrich Gauss gave over a whole chapter of his great 1801 classic Disquisitiones Arithmeticae to it—54 pages in the English translation.
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Root of Unity -- from Wolfram MathWorld
th root of unity. The roots of unity. +1 is always an n th root of unity, but -1 is such a root only ...
[9]
[PDF] Caltech Harvey Mudd Mathematics Competition
Nov 23, 2013 · By Vieta's formula, the product of the roots is given by (−1)n −1 1 = (−1)n+1. So the product of the roots of unity is 1 when n is odd and −1 ...
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[PDF] Fields and Cyclotomic Polynomials
Definition: Root of Unity. If n is a positive integer, an nth root of unity is a complex number ζ such that. ζn = 1.Missing: mathematics | Show results with:mathematics
[11]
Primitive Roots of Unity | Brilliant Math & Science Wiki
Basic Properties. The primitive n th n^\text{th} nth roots of unity are the complex numbers. e 2 π i k / n : 1 ≤ k ≤ n , gcd ( k , n ) = 1. e^{2\pi i k/n} ...
[12]
[PDF] Chapter 13: Complex Numbers - Sections 13.1 & 13.2 - Arizona Math
Euler's formula reads exp(iθ) = cos(θ) + i sin(θ), θ ∈ R. (). Chapter 13 ... The n-th roots of unity are given by n. √. 1 = cos. 2kπ n. + i sin. 2kπ n. = ωk ...
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[PDF] Complex Numbers
has n solutions, which are called nth roots of unity. To find them, we use. Euler's formula (4.5) to observe that, for any integer k, e2πki = 1. (5.2). Thus ...
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[PDF] 4. Roots of unity Theorem 4.1 (De Moivre's Theorem). (cosθ + isinθ ...
Equating real and imaginary parts we get cos 3θ = cos3 θ - 3 cosθ sin2 θ sin 3θ = 3 cos2 θ sinθ - sin3 θ. We can also use Euler's formula to compute nth roots.
[15]
Complex Number Primer - Pauls Online Math Notes
We now need to move onto computing roots of complex numbers. We'll start this off “simple” by finding the nth roots of unity. The nth roots of unity for n=2,3,…
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[PDF] Constructible Regular n-gons
May 8, 2013 · Note that the n-th roots of unity are n evenly spaced points around the complex unit circle that when connected form a regular n-gon and that ...
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[PDF] 19. Roots of unity
One of the general uses of Galois theory is to understand fields intermediate between a base field k and an algebraic field extension K of k. In the case of ...
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root of unity in nLab
Mar 23, 2025 · An n n th root of unity in a ring R R is an element x x such that x n = 1 x^n = 1 in R R , hence is a root of the equation x n − 1 = 0 x^n-1 = 0 ...
[19]
[PDF] Roots of Unity
Q[ζn]Q[ζm] is the cyclotomic extension generated by the primitive mnth root of unity ζnζm, of degree φ(mn) = φ(m)φ(n) over Q. 2. Now look at the tower L = L0 ...Missing: definition | Show results with:definition
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[PDF] Homework # 9 Math 3340 Spring 2021 - Cornell Mathematics
. (c) Show that cos(2π/5) = −1+. √. 5. 4. , and that sin(2π/5) = √. 10+2. √. 5. 4 . (d) Conclude that we can construct a regular pentagon. Challenge, not for HW ...Missing: exact | Show results with:exact
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(PDF) Solving Cyclotomic Polynomials by Radical Expressions
We describe a Maple package that allows the solution of cyclotomic polynomials by radical expressions. We provide a function that is an extension of the ...
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[PDF] galois theory and the abel-ruffini theorem
field F which contains the nth roots of unity is of the form F( n. √ a) for some a ∈ F. Definition 5.24. An element α that is algebraic over F can be expressed ...
[23]
Computing radical expressions for roots of unity - ACM Digital Library
With ~p we will denote a primitive p-th root of unity. Having a fixed p-th root of unity ¢ and some primitive root rood p--which will be denoted by g--GAUSS ...Missing: fifth | Show results with:fifth
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[PDF] Section 1.9. Roots of Complex Numbers
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A cyclotomic extension of K is a field K(ζn) where ζn is a root of unity (of order n). The term cyclotomic means circle-dividing.
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[PDF] Algebra - UT Math
Consider the group G of all the roots of unity in C. First note that G is isomorphic to Q/Z. This is given by m n. 7→ exp (2πim/n) Then we have that. G := {z ...
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[PDF] (b) All the 16th roots of -5. Solution
Let S be the set of all roots of unity: S = 1z ∈ C : zn = 1 for some n ∈ Z+l. Show that S is dense in the unit circle S1, i.e., for every w ∈ S1 and every > 0, ...
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[PDF] MAT313 Fall 2017 Practice Midterm II
The image coincides with group of roots of unity. The kernel is the set of integers. By the first isomorphism theorem Q/Z is isomorphic to the group of unity, ...
[29]
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### Summary of Q/Z Structure and Relation to Roots of Unity
[30]
None
### Summary of Primitive nth Roots of Unity from the Document
[31]
[PDF] math 361: number theory — third lecture
The relation f ∗ u = g = id says that f is the convolution inverse of u, the Möbius function. That is, sum of the primitive nth roots of 1 = µ(n).
[32]
[PDF] 8. Cyclotomic polynomials
Since P is irreducible, either e − 1=0in k, or P divides. R, or P divides DP. If P divides R, Pe divides f, and we're done. If P does not divide R then P ...
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[PDF] Several Proofs of the Irreducibility of the Cyclotomic Polynomial.
THE RESULTS THEMSELVES Theorem 1. Let p be a prime. Then the cyclotomic polynomial Φp(x) is irreducible. Proof (Gauss). This is trivial for p = 2 so we suppose ...
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[PDF] Fields and Galois Theory - James Milne
These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental ...Missing: ramification | Show results with:ramification
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[PDF] theorem 1: proof for cyclotomic extensions
Let n be a positive rational integer and ζn a primitive n-th root of unity. Rational prime p is ramified in Q(ζn)/Q only if p divides n. Proof. Let O be the ...
[36]
Constructible Polygon -- from Wolfram MathWorld
... Gauss gave a sufficient condition for a regular n -gon to be constructible, which he also conjectured (but did not prove) to be necessary, thus showing that ...
[37]
Ramanujan's Sum -- from Wolfram MathWorld
The sum c_q(m)=sum_(h^*(q))e^(2piihm/q), where h runs through the residues relatively prime to q, which is important in the representation of numbers by the ...Missing: formula | Show results with:formula
[38]
[PDF] A Course in Finite Group Representation Theory
3.2 Orthogonality relations and bilinear forms . ... roots of unity, and roots of unity (in some extension ring if necessary) ...
[39]
[PDF] FINITE FOURIER ANALYSIS 1. The group Z Let N be a positive ...
We denote the set of all Nth roots of unity by ZN . It is an easy exercise (do it!) to see that. ZN is an abelian group under complex multiplication. It turns ...
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Linear Difference Equations with Discrete Transform Methods
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The aim of this handout is to show that Gal(Kn/Q) → (Z/nZ)× is an isomorphism even when n is not a prime power. We cannot hope to use Eisenstein's ...
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[PDF] Class Field Theory
Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself.
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[PDF] A Fast Multiplication Algorithm and RLWE–PLWE Equivalence for ...
These number fields are generated by the real number ψn = 2 cos(2π/n) and the degree of the extension over Q is φ(n)/2. In addition, these fields are the fixed ...
[44]
Chebyshev Polynomials and the Minimal Polynomial of cos (2π/n)
Aug 6, 2025 · The minimal polynomial of cos (2π/n) allows one to realize the value of cos (2π/n) as the root of a polynomial with rational coefficients.
[45]
[PDF] On the class number formula of certain real quadratic fields - HAL
Feb 3, 2015 · 0<v<d/2. 2 cos 2π nv. 2 o. The remaining part is the same as in [3] and ... 1 = T −2 (which follows from the Pell equation T2. 1 −U2. 1 d ...