Properties
A cyclotomic field is the splitting field of the polynomial
- xn − 1
and therefore it is a Galois extension of the field of rational numbers. The degree of the extension
is given by φ(n) where φ is Euler's phi function. A complete set of Galois conjugates is given by { (ζn)a } , where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group is naturally isomorphic to the multiplicative group
- (Z/nZ)×
of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula
- b: (ζn)a → (ζn)a b.
Read more about this topic: Cyclotomic Field
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