Cyclotomic field: Difference between revisions
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In [[number theory]], a '''cyclotomic field''' is a [[number field]] which is obtained by adjoining a [[root of unity]] to '''Q'''. The ''n''-th cyclotomic field is obtained by adjoining a primitive ''n''-th root of unity ''ζ''<sub>n</sub> and denoted by '''Q'''(''ζ''<sub>n</sub>). It's the splitting field of the polynomial ''x''<sup>n</sup> - 1 and therefore it's a [[Galois extension]]. Its [[Galois group]] is naturally isomorphic to the multiplicative group ('''Z'''/n'''Z''')<sup>x</sup>. The degree of the extension ['''Q'''(''ζ''<sub>n</sub>):'''Q'''] is given by ''φ''(''n'') where φ is [[Euler's phi function]]. |
In [[number theory]], a '''cyclotomic field''' is a [[number field]] which is obtained by adjoining a [[root of unity]] to '''Q'''. The ''n''-th cyclotomic field is obtained by adjoining a primitive ''n''-th root of unity ''ζ''<sub>n</sub> and denoted by '''Q'''(''ζ''<sub>n</sub>). It's the splitting field of the polynomial ''x''<sup>n</sup> - 1 and therefore it's a [[Galois extension]]. Its [[Galois group]] is naturally isomorphic to the multiplicative group ('''Z'''/n'''Z''')<sup>x</sup>. The degree of the extension ['''Q'''(''ζ''<sub>n</sub>):'''Q'''] is given by ''φ''(''n'') where φ is [[Euler's phi function]]. |
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==Reference== |
==Reference== |
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Revision as of 23:12, 6 April 2007
In number theory, a cyclotomic field is a number field which is obtained by adjoining a root of unity to Q. The n-th cyclotomic field is obtained by adjoining a primitive n-th root of unity ζn and denoted by Q(ζn). It's the splitting field of the polynomial xn - 1 and therefore it's a Galois extension. Its Galois group is naturally isomorphic to the multiplicative group (Z/nZ)x. The degree of the extension [Q(ζn):Q] is given by φ(n) where φ is Euler's phi function.
Reference
- Daniel A. Marcus, Number Fields, third edition, Springer-Verlag, 1977