Root of unity | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field. An nth root of unity, where n is a positive integer, is a number z satisfying the equation Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if n is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are However, the defining equation of roots of unity is meaningful over any field (and even over any ring) F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details. An nth root of unity is said to be if it is not an mth root of unity for some smaller m, that is if If n is a prime number, then all nth roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (171)

Related courses (23)

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-317: Galois theory

Ce cours commence par un rappel de la théorie de Galois vue en 2ème année : les extensions de corps et la correspondance de Galois. Ensuite, on présente les applications et approfondissements.

ME-201: Continuum mechanics

Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and de

Related lectures (207)

Discriminant

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial is the quantity which appears under the square root in the quadratic formula.

Nth root

In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as , where x is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition then of an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: A root of degree 2 is called a square root (usually written without the n as just ) and a root of degree 3, a cube root (written ).

Root of unity

Calcul de valeurs propres MATH-212: Analyse numérique et optimisation

Covers the calculation of eigenvalues and eigenvectors, emphasizing their significance and applications.

Complex Numbers: Equations and Constructibility MATH-110(a): Advanced linear algebra I

Explores polynomial equations in complex numbers, constructibility conditions, roots of unity, and Fermat primes.

Sets and Relations MATH-101(g): Analysis I

Introduces sets, Cartesian product, order importance, and relations on sets.