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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 ...
over the field , then the Galois group of the polynomial is defined as the ... where is a primitive cube root of unity. The group (/) is isomorphic to ...
If a factor group in the composition series is cyclic of order n, and if in the corresponding field extension L/K the field K already contains a primitive n th root of unity, then it is a radical extension and the elements of L can then be expressed using the n th root of some element of K.
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 ...
The proof is based on the fundamental theorem of Galois theory and the following theorem. Let K be a field containing n distinct n th roots of unity. An extension of K of degree n is a radical extension generated by an nth root of an element of K if and only if it is a Galois extension whose Galois group is a cyclic group of order n.
In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → Aut R (R(1)) ≈ GL(1, R)).
The units of a Galois ring R are all the elements which are not multiples of p. The group of units, R ×, can be decomposed as a direct product G 1 ×G 2, as follows. The subgroup G 1 is the group of (p r – 1)-th roots of unity. It is a cyclic group of order p r – 1. The subgroup G 2 is 1+pR, consisting of all elements congruent to 1 modulo p.
The group of roots of unity in Q(ζ n) has order n or 2n, according to whether n is even or odd. The unit group Z [ζ n ] × is a finitely generated abelian group of rank φ ( n )/2 – 1 , for any n > 2 , by the Dirichlet unit theorem .