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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.
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".
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.
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
This is a finite field, and primitive n th roots of unity exist whenever n divides , so we have = + for a positive integer ξ. Specifically, let ω {\displaystyle \omega } be a primitive ( p − 1 ) {\displaystyle (p-1)} th root of unity, then an n th root of unity α {\displaystyle \alpha } can be found by letting α = ω ξ {\displaystyle ...
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.