Cyclotomic field: Difference between revisions

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In [[mathematics]], a '''cyclotomic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] generated by [[root of unity|roots of unity]].  If ζ denotes an ''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>.
In [[mathematics]], a '''cyclotomic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] generated by [[root of unity|roots of unity]].  If ζ denotes an ''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>.



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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