Cyclotomic field: Difference between revisions

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==Splitting of primes==
==Splitting of primes==
A prime ''p'' [[ramification|ramifies]] iff ''p'' divides ''n''.  Otherwise, the splitting of ''p'' depends on the common factor of ''p''-1 and ''n''.
A prime ''p'' [[ramification|ramifies]] iff ''p'' divides ''n''.  Otherwise, the splitting of ''p'' depends on the factorisation of the polynomial <math>X^n-1</math> modulo ''p'', which in turn depends on the [[highest common factor]] of ''p''-1 and ''n''.
 
==Galois group==
The [[minimal polynomial]] for ζ is the ''n''-th [[cyclotomic polynomial]] <math>\Phi_n(X)</math>, which is a factor of <math>X^n-1</math>.  Since the powers of ζ are the roots of the latter polynomial, ''F'' is a [[splitting field]] for <math>\Phi_n(X)</math> and hence a [[Galois extension]].  The [[Galois group]] is [[group isomorphism|isomorphic]] to the [[multiplicative group]], <math>(\mathbf{Z}/n\mathbf{Z})^*</math> via
 
:<math>a \bmod n \leftrightarrow \sigma_a = (\zeta\mapsto\zeta^a) .\,</math>


==References==
==References==

Revision as of 13:36, 7 December 2008

In mathematics, a cyclotomic field is a field which is an extension generated by roots of unity. If ζ denotes an n-th root of unity, then the n-th cyclotomic field F is the field extension .

Ring of integers

As above, we take ζ to denote an n-th root of unity. The maximal order of F is

Unit group

Class group

Splitting of primes

A prime p ramifies iff p divides n. Otherwise, the splitting of p depends on the factorisation of the polynomial modulo p, which in turn depends on the highest common factor of p-1 and n.

Galois group

The minimal polynomial for ζ is the n-th cyclotomic polynomial , which is a factor of . Since the powers of ζ are the roots of the latter polynomial, F is a splitting field for and hence a Galois extension. The Galois group is isomorphic to the multiplicative group, via

References