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For n = 1, the cyclotomic polynomial is Φ 1 (x) = x − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive n th root of unity for every n > 1. As Φ 2 (x) = x + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive n th root of unity for every even n > 2.
A non-example is in the ring of integers modulo ; while () and thus is a cube root of unity, + + meaning that it is not a principal cube root of unity. The significance of a root of unity being principal is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is
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 x to the equation (or congruence) (). If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. [1]
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
As the 3rd and the 7th roots of unity belong to GF(4) and GF(8), respectively, the 54 generators are primitive n th roots of unity for some n in {9, 21, 63}. Euler's totient function shows that there are 6 primitive 9 th roots of unity, 12 {\displaystyle 12} primitive 21 {\displaystyle 21} st roots of unity, and 36 {\displaystyle 36} primitive ...
However, there are no primitive 3rd roots of unity in a real closed field. Suppose that ω is a primitive 3rd root of unity. Then, by the axioms defining an ordered field , ω and ω 2 are both positive, because otherwise their cube (=1) would be negative.
is a primitive (hence non-real) cube root of unity. Eisenstein integers as the points of a certain triangular lattice in the complex plane. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably ...