Cyclic group
Galois theory

An n th root of unity is a complex number whose n th power is 1, a root of the polynomial x − 1. The set of all n th roots of unity forms a cyclic group of order n under multiplication. The generators of this cyclic group are the n th primitive roots of unity; they are the roots of the n th cyclotomic polynomial.For example, the polynomial z − 1 factors as (z − 1)(z − ω)(z − ω), where ω = e; the set {1, ω, ω } = { ω, ω, ω } forms a cyclic group under multiplication. The Galois group of the field extension of the rational numbers generated by the n th roots of unity forms a different group, isomorphic to the multiplicative group (Z/ n Z) of order φ (n), which is cyclic for some but not all n (see above).

A field extension is called a cyclic extension if its Galois group is cyclic. For fields of characteristic zero, such extensions are the subject of Kummer theory, and are intimately related to solvability by radicals. For an extension of finite fields of characteristic p, its Galois group is always finite and cyclic, generated by a power of the Frobenius mapping. Conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G.
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