In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor...

geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle)...

by ζ n {\displaystyle \zeta _{n}} . The n {\displaystyle n} th cyclotomic polynomial Φ n ( x ) = ∏ gcd ( k , n ) = 1 1 ≤ k ≤ n ( x − e 2 π i k / n )...

important class of polynomials whose irreducibility can be established using Eisenstein's criterion is that of the cyclotomic polynomials for prime numbers...

unity. Then the minimal polynomial of ζ n {\displaystyle \zeta _{n}} is given by the n {\displaystyle n} -th cyclotomic polynomial Φ n ( x ) {\displaystyle...

denominator, 1 − z j, which is the product of cyclotomic polynomials. The left hand side of the cyclotomic identity is the generating function for the free...

irreducible cyclotomic polynomials", Electronics and Communications in Japan, 74 (4): 106–113, doi:10.1002/ecjc.4430740412, MR 1136200. all one polynomial at PlanetMath...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

All-one polynomials Abel polynomials Bell polynomials Bernoulli polynomials Cyclotomic polynomials Dickson polynomials Fibonacci polynomials Lagrange...

of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization...

those associated with the cyclotomic polynomials of degrees 5 and 17. Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable...

of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials. If a is a root of a polynomial that is either palindromic or antipalindromic...

evidence it is not known that this sequence extends indefinitely. The cyclotomic polynomials Φ k ( x ) {\displaystyle \Phi _{k}(x)} for k = 1 , 2 , 3 , … {\displaystyle...

{\displaystyle n} th cyclotomic polynomial are distinct in every field of characteristic p {\displaystyle p} , as this polynomial is a divisor of X n −...

class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials Φ n {\displaystyle \Phi _{n}} defined as Φ n ( x ) =...

not divide n, and Q n ( x ) {\displaystyle Q_{n}(x)} is the nth cyclotomic polynomial. For example, a 6 − b 6 = Q ¯ 1 ( a , b ) Q ¯ 2 ( a , b ) Q ¯ 3...

2, the polynomial Φ(x) will be the cyclotomic polynomial xn + 1. Other choices of n are possible but the corresponding cyclotomic polynomials are more...

function φ(n) in number theory; also called Euler's phi function. The cyclotomic polynomial functions Φn(x) of algebra. The number of electrical phases in a...

{\displaystyle x^{n}-1=0} for any n, and the equations defined by cyclotomic polynomials, all of whose solutions can be expressed in radicals. Abel's proof...

{Q} )} . The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials. The roots of the minimal polynomial of 2cos(2π/n) are twice...

the fourth cyclotomic polynomial. As with the cyclotomic polynomials more generally, Φ 4 {\displaystyle \Phi _{4}} is an irreducible polynomial, so this...

Equivalently, a regular n-gon is constructible if any root of the nth cyclotomic polynomial is constructible. Restating the Gauss–Wantzel theorem: A regular...

{\displaystyle \Phi _{d}(x)} is the d t h {\displaystyle d^{\mathrm {th} }} cyclotomic polynomial and d ranges over the divisors of n. For p prime, Φ p ( x ) = ∑...

prime Emma Lehmer, "On the magnitude of the coefficients of the cyclotomic polynomial", Bulletin of the American Mathematical Society 42 (1936), no. 6...

mathematician and educator. Her research focused on the area of cyclotomic polynomials. Beiter was born in Buffalo to Kathryn (née Kiel) and Edward Frederick...

measure of an irreducible non-cyclotomic polynomial. Lehmer's polynomial is a factor of the shorter degree-12 polynomial, Q ( x ) = x 12 − x 7 − x 6 −...

primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published...

has a prime factor p such that any kth cyclotomic polynomial Φk(p) is smooth. The first few cyclotomic polynomials are given by the sequence Φ1(p) = p−1...

identity for polynomials. x n − 1 = ∏ d | n Φ d ( x ) {\displaystyle x^{n}-1=\prod _{d|n}\Phi _{d}(x)} , a product of cyclotomic polynomials. Factoring...

Brahmagupta polynomials Caloric polynomial Charlier polynomials Chebyshev polynomials Chihara–Ismail polynomials Cyclotomic polynomials Dickson polynomial Ehrhart...