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The n th roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of ...
The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n.
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.
K contains n distinct nth roots of unity (i.e., roots of X n − 1) L/K has abelian Galois group of exponent n. For example, when n = 2, the first condition is always true if K has characteristic ≠ 2. The Kummer extensions in this case include quadratic extensions = where a in K is a non-square element.
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a 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 α i for some natural number i.
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 ...
The Chebotarev theorem on roots of unity was originally a conjecture made by Ostrowski in the context of lacunary series.. Chebotarev was the first to prove it, in the 1930s. . This proof involves tools from Galois theory and pleased Ostrowski, who made comments arguing that it "does meet the requirements of mathematical esthetics".
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