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 Fields
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)