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...

mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic Primitive nth root of unity amongst the solutions of zn = 1 in a field...

In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element α {\displaystyle \alpha } satisfying the equations...

determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman...

primitive root modulo n (or in fuller language primitive root of unity modulo n, emphasizing its role as a fundamental solution of the roots of unity polynomial...

principal nth root of unity, defined by: The discrete Fourier transform maps an n-tuple ( v 0 , … , v n − 1 ) {\displaystyle (v_{0},\ldots ,v_{n-1})} of elements...

series of genealogy programs nth root of a number Root of unity, a complex number which is an nth root of one Root of an equation, a solution of the equation...

Heckenberger: Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis 2005. Heckenberger, Schneider: Root system and Weyl gruppoid...

In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution...

number of nth roots of unity in GF(q) is gcd(n, q − 1). In a field of characteristic p, every (np)th root of unity is also a nth root of unity. It follows...

mathematician and engineer Heron of Alexandria is noted as the first to present a calculation involving the square root of a negative number, it was Rafael...

α and all of its conjugates in the complex numbers have absolute value at most 1, then α is a root of unity. There are quantitative forms of this, stating...

the field of the rational numbers of any primitive nth-root of unity ( e 2 i π / n {\displaystyle e^{2i\pi /n}} is an example of such a root). An important...

complex root of unity to Q {\displaystyle \mathbb {Q} } , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern...

always abelian. If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting Kummer extension is an...

changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a primitive cube root of unity, that is ⁠–1 ± √–3/2⁠...

Methods of computing square roots List of polynomial topics Nth root Square root Nested radical Root of unity "In Search of a Fast Cube Root". metamerist...

(disambiguation), one of two concepts Primitive function or antiderivative, F′ = f Primitive permutation group Primitive root of unity; See Root of unity Primitive...

root of unity with the smallest positive argument, it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of...

above. Apotome (mathematics) Cube root Functional square root Integer square root Nested radical Nth root Root of unity Solving quadratic equations with...

exactly the elements of Cp of the form pr·ζ where r is a rational number and ζ is a root of unity. Note that there is no analogue in Cp of Euler's identity...

cyclic (see Root of unity § Cyclic groups). In addition to the multiplication of two elements of F, it is possible to define the product n ⋅ a of an arbitrary...

primitive element if it is a primitive (q − 1)th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as αi for some natural...

the entire field GF(pm). This implies that α is a primitive (pm − 1)-root of unity in GF(pm). Because all minimal polynomials are irreducible, all primitive...

nth root of unity. Then the n-torsion on E ( K ¯ ) {\displaystyle E({\overline {K}})} is known to be a Cartesian product of two cyclic groups of order...

primitive root of unity. The field Q ( 2 3 , ζ 3 ) {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}},\zeta _{3})} is the normal closure (see below) of Q ( 2 3...

orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity): ∑ m = 0...

that e − 2 π i / n {\textstyle e^{-2\pi i/n}} is an n'th primitive root of unity, and thus can be applied to analogous transforms over any finite field...

primitive p {\displaystyle p} -th root of unity. Indeed, if x {\displaystyle x} is a p {\displaystyle p} -th root of unity in k {\displaystyle k} , then it...

D {\displaystyle D} th root of unity modulo 2 n ′ + 1 {\displaystyle 2^{n'}+1} . We now take the discrete Fourier transform of the arrays A , B {\displaystyle...