Abelian group
TheoremComplex roots of unity#
The complex roots of unity are the complex numbers \(z\) that satisfy \(z^n = 1\) for some positive integer \(n\). For each \(n \ge 1\) the equation \(z^n = 1\) has exactly \(n\) solutions, the numbers \(e^{2\pi i k/n}\) with \(0 \le k < n\), and they form a cyclic group of order \(n\) under multiplication. They are the standard model of a finite cyclic group, they describe the regular polygons inscribed in the unit circle, and they are the roots of the cyclotomic polynomials, the values of characters of finite abelian groups and the sample points of the discrete Fourier transform.
Definition
Let \(n\) be a positive integer. A complex number \(z\) is an \(n\)-th root of unity if \(z^n = 1\). Such a \(z\) is nonzero, so it belongs to the multiplicative group \(\mathbb{C}^\times\) of nonzero complex numbers, and the \(n\)-th roots of unity form the subgroup of \(\mathbb{C}^\times\) of elements whose \(n\)-th power is \(1\), usually written \(\mu_n\): a unit lies in it exactly when its \(n\)-th power is \(1\). It is a subgroup because \(1^n = 1\), \((zw)^n = z^n w^n\) and \((z^{-1})^n = (z^n)^{-1}\).
Statement
For every positive integer \(n\), a nonzero complex number \(z\) is an \(n\)-th root of unity if and only if \(z = e^{2\pi i k/n}\) for some integer \(k\) with \(0 \le k < n\). These \(n\) numbers are pairwise distinct, so \(\mu_n\) has exactly \(n\) elements. In terms of the trigonometric functions,
so for \(n \ge 3\) the \(n\)-th roots of unity are the vertices of the regular \(n\)-gon inscribed in the unit circle with one vertex at \(1\).
Proof
The main tool is the complex exponential and the fact that \(e^w = 1\) exactly when \(w\) is an integer multiple of \(2\pi i\).
If \(z = e^{2\pi i k/n}\), then, since \(e^{mw} = (e^w)^m\) for every natural number \(m\), \(z^n = e^{2\pi i k} = 1\). Conversely, suppose \(z^n = 1\). Taking absolute values gives \(|z|^n = 1\), so \(|z| = 1\), and \(z = e^{i\theta}\) for some real \(\theta\). Then \(e^{in\theta} = 1\), so \(n\theta = 2\pi m\) for an integer \(m\). Dividing \(m\) by \(n\) with remainder, \(m = qn + k\) with \(0 \le k < n\), gives \(z = e^{2\pi i q}\, e^{2\pi i k/n} = e^{2\pi i k/n}\).
For distinctness, if \(e^{2\pi i k/n} = e^{2\pi i l/n}\) with \(0 \le l \le k < n\), then \(e^{2\pi i (k-l)/n} = 1\), so \((k-l)/n\) is an integer in \([0, 1)\), hence \(k = l\).
Examples
- For \(n = 1\) the only root is \(1\), and for \(n = 2\) the roots are \(1\) and \(-1\).
- For \(n = 3\) the roots are \(1\), \(\omega = -\tfrac12 + \tfrac{\sqrt3}{2} i\) and \(\omega^2 = \bar\omega = -\tfrac12 - \tfrac{\sqrt3}{2} i\).
- For \(n = 4\) the roots are \(1, i, -1, -i\); the powers of \(i\) run through all four, and \(i\) is a primitive fourth root of unity.
Properties
- Cyclic structure. The number \(\omega_n = e^{2\pi i/n}\) is a primitive \(n\)-th root of unity: its powers \(1, \omega_n, \dots, \omega_n^{n-1}\) are exactly the elements of \(\mu_n\), so \(\mu_n\) is cyclic of order \(n\), and the order of \(\omega_n^k\) is \(n/\gcd(k, n)\). More generally, for \(n \ge 1\), the \(n\)-th roots of unity in any integral domain form a cyclic group.
- Primitive roots. The root \(e^{2\pi i k/n}\) generates \(\mu_n\) exactly when \(\gcd(k, n) = 1\) (see primitive complex roots of unity), so there are \(\varphi(n)\) primitive \(n\)-th roots of unity, where \(\varphi\) is Euler's totient function.
- Unit circle and conjugation. For \(n \ge 1\), every \(n\)-th root of unity has absolute value \(1\), and so its complex conjugate is its inverse.
- Divisibility. If \(d\) divides \(n\), then every \(d\)-th root of unity is an \(n\)-th root of unity.
- Vanishing sum. If \(\omega\) is a primitive \(n\)-th root of unity and \(n \ge 2\), then \(1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0\), so the \(n\)-th roots of unity add up to \(0\).
- Factorization. For \(n \ge 1\), since \(\mathbb{C}\) contains a primitive \(n\)-th root of unity, \(X^n - 1\) is the product of the factors \(X - \zeta\) over all \(\zeta \in \mu_n\). Grouping the roots by their order shows that \(X^n - 1\) is the product of the cyclotomic polynomials \(\Phi_d\) over the divisors \(d\) of \(n\) (see cyclotomic polynomial).
Related concepts
Every finite subgroup \(G\) of \(\mathbb{C}^\times\) of order \(n\) equals \(\mu_n\). For each \(g \in G\), Lagrange's theorem applied to the cyclic subgroup generated by \(g\) shows that the order of \(g\) divides \(n\), so \(g^n = 1\). Hence \(G \subseteq \mu_n\), and both sets have \(n\) elements. An element of \(\mathbb{C}^\times\) has finite order exactly when it is a root of unity. A homomorphism from a finite group of order \(n\) to \(\mathbb{C}^\times\) takes its values in \(\mu_n\), which is why roots of unity appear throughout the theory of dual groups and the Fourier transform for finite abelian groups.
In Mathlib
Mathlib defines the \(n\)-th roots of unity for any commutative monoid, as a subgroup of its group of units, and the membership criterion holds in that generality. For the complex numbers the units are the nonzero complex numbers, so a complex root of unity is a nonzero complex number viewed as a unit. The exponent is a natural number and may be \(0\); since \(z^0 = 1\), the \(0\)-th roots of unity are all the units, so the explicit description, the primitive root \(e^{2\pi i/n}\), the count, cyclicity and the absolute-value and conjugation statements carry the standing hypothesis that \(n\) is nonzero. The explicit description writes the exponent as \(2\pi i \cdot (k/n)\) with \(k\) a natural number below \(n\). The number of elements is measured by the natural-number cardinality. Primitive roots are a separate predicate on elements: \(\zeta\) is a primitive \(n\)-th root of unity when \(\zeta^n = 1\) and \(n\) divides every exponent \(l\) with \(\zeta^l = 1\). Cyclicity, the vanishing sum and the factorization of \(X^n - 1\) into linear factors (given a primitive \(n\)-th root) are proved over arbitrary integral domains, and the factorization into cyclotomic polynomials over any commutative ring.
Written for Trough, under CC BY-SA 4.0.
Sources
Uses 3 · Used by 1
TheoremPrimitive complex roots of unity#
For a positive integer \(n\), a primitive \(n\)-th root of unity is a complex number \(\zeta\) with \(\zeta^n = 1\) whose powers \(\zeta, \zeta^2, \dots, \zeta^{n-1}\) all differ from \(1\). The primitive \(n\)-th roots are the generators of the cyclic group of complex \(n\)-th roots of unity, and they have an explicit description: they are the numbers \(e^{2\pi i k/n}\) with \(k\) coprime to \(n\). They are the roots of the cyclotomic polynomials, and they appear throughout algebra and number theory, from the structure of finite cyclic groups to Fourier analysis on them.
Definition
Let \(n \ge 1\). A complex number \(\zeta\) is a primitive \(n\)-th root of unity when \(\zeta^n = 1\) and \(\zeta^m \neq 1\) for every \(m\) with \(0 < m < n\). Equivalently, \(\zeta\) has order exactly \(n\) in the multiplicative group of nonzero complex numbers: \(\zeta\) is a primitive \(n\)-th root precisely when its order is \(n\).
Statement
For \(n \ge 1\), the primitive \(n\)-th roots of unity in \(\mathbb{C}\) are exactly the numbers \(e^{2\pi i k/n}\) with \(0 \le k < n\) and \(\gcd(k, n) = 1\). These numbers lie on the unit circle at the angles \(2\pi k/n\), and distinct values of \(k\) in this range give distinct numbers, so there are exactly \(\varphi(n)\) primitive \(n\)-th roots of unity, where \(\varphi\) is Euler's totient function.
Examples
- For \(n = 1\) the only primitive root is \(1\) itself, and for \(n = 2\) it is \(-1\).
- For \(n = 4\) the primitive roots are \(i\) and \(-i\): \(i\) is a primitive fourth root of unity, and so is \(-i\), while \(-1\) is not, since \((-1)^2 = 1\).
- For \(n = 3\) the primitive roots are \(\tfrac{-1 \pm i\sqrt{3}}{2}\), and for \(n = 6\) they are \(\tfrac{1 \pm i\sqrt{3}}{2}\).
- For every \(n \ge 1\), the number \(e^{2\pi i/n}\) is a primitive \(n\)-th root of unity.
- More generally, if \(q = a/b\) is a rational number in lowest terms with \(b \ge 1\), then \(e^{2\pi i q}\) is a primitive \(b\)-th root of unity.
Proof
Write \(\omega = e^{2\pi i/n}\). By the periodicity of the complex exponential, \(\omega^m = e^{2\pi i m/n}\) equals \(1\) exactly when \(n\) divides \(m\), so \(\omega\) has order \(n\). The \(n\) numbers \(1, \omega, \dots, \omega^{n-1}\) are distinct roots of \(x^n - 1\), a polynomial with at most \(n\) roots, so every \(n\)-th root of unity is \(\omega^k = e^{2\pi i k/n}\) for a unique \(k\) with \(0 \le k < n\). Finally, the order of \(\omega^k\) is \(n/\gcd(k, n)\), which equals \(n\) exactly when \(\gcd(k, n) = 1\).
Properties
Throughout, \(n \ge 1\) and \(\zeta\) is a primitive \(n\)-th root of unity.
- Generators. Every \(n\)-th root of unity is a power of \(\zeta\): each \(\xi\) with \(\xi^n = 1\) equals \(\zeta^k\) for some \(k\) with \(0 \le k < n\). So the \(n\)-th roots of unity form a cyclic group of order \(n\), and its generators are exactly the primitive \(n\)-th roots of unity.
- Powers. The power \(\zeta^k\) is a primitive \(n\)-th root exactly when \(\gcd(k, n) = 1\), and in general \(\zeta^k\) is a primitive \((n/\gcd(n, k))\)-th root of unity. Thus the primitive \(n\)-th roots correspond to the invertible residues modulo \(n\).
- Absolute value. Every primitive \(n\)-th root has absolute value \(1\).
- Vanishing sum. For \(n \ge 2\), \(1 + \zeta + \zeta^2 + \cdots + \zeta^{n-1} = 0\), because the sum is a geometric series and \(\zeta^n = 1\) while \(\zeta \neq 1\).
- Sorting the roots of unity by order. A complex number is a primitive root of unity for at most one exponent: if \(\zeta\) is a primitive \(k\)-th root and a primitive \(l\)-th root, then \(k = l\). Each \(n\)-th root of unity has some order \(d\) dividing \(n\), so the \(n\)-th roots of unity are the union, over the divisors \(d\) of \(n\), of the primitive \(d\)-th roots. Counting both sides gives the identity \(\sum_{d \mid n} \varphi(d) = n\).
Cyclotomic polynomials
The \(n\)-th cyclotomic polynomial \(\Phi_n\) collects the primitive \(n\)-th roots. In any integral domain that contains a primitive \(n\)-th root of unity, such as \(\mathbb{C}\), the polynomial \(\Phi_n\) is the product of \(x - \zeta\) over the primitive \(n\)-th roots \(\zeta\), and for \(n \ge 1\) a complex number is a root of \(\Phi_n\) exactly when it is a primitive \(n\)-th root of unity. Its degree is \(\varphi(n)\). Its coefficients are integers: over any ring, \(\Phi_n\) is the image of the integer polynomial \(\Phi_n\), and for \(n \ge 1\) it is irreducible over the rationals, so all primitive \(n\)-th roots are conjugate algebraic numbers.
In Mathlib
Mathlib defines primitive roots of unity for an element of any commutative monoid, not only for complex numbers: \(\zeta\) is a primitive \(n\)-th root when \(\zeta^n = 1\) and \(n\) divides every exponent \(m\) with \(\zeta^m = 1\). This agrees with the definition above for \(n \ge 1\), but it also gives a meaning to \(n = 0\): a primitive \(0\)-th root is an element none of whose positive powers is \(1\). In \(\mathbb{C}\) every number that is not a root of unity, \(0\) and \(2\) among them, is a primitive \(0\)-th root, which is why the description by exponentials assumes \(n \neq 0\). This matches the convention that an element of infinite order has order \(0\). The finite set of primitive \(n\)-th roots is taken to be empty for \(n = 0\), consistent with \(\varphi(0) = 0\), so the count of primitive roots holds for every \(n\). The theorems on generators, vanishing sums, the decomposition by divisors, and the product and root descriptions of \(\Phi_n\) are proved in any integral domain, not only in \(\mathbb{C}\); there the root criterion for \(\Phi_n\) asks that \(n\) be nonzero as an element of the domain. The lemmas on powers of a primitive root hold in any commutative monoid, the degree formula holds over any nontrivial ring, and irreducibility is stated over \(\mathbb{Q}\).
Written for Trough, under CC BY-SA 4.0.
Sources
Uses 6
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