Cyclotomic Fields [HB 55]

The cyclotomic fields have been rewritten to become a part of the algebraic fields and their orders. Cyclotomic Fields (FldCyc) now inherit from the Number Fields (FldNum). This has resulted in increased speed for most operations and extended the number of available operations greatly.

Removals and Changes:

• Conjugate(a, n) now returns a conjugate of a. The signature Conjugate(a, r) should be used to retrieve the conjugate of a under the automorphism $\zeta_m \mapsto \zeta_m^n$ where r is $\zeta_m^n$ for some n coprime to m, the order of the field.

• The elt constructor now takes as many coefficients as the degree of the field instead of an arbitrary number. Coercion will only take sequences of length the degree of the field.

New Features:

• Orders of cyclotomic fields can be found from the field.

• Ideals and quotients of cyclotomic orders can be formed.

• Cyclotomic fields and orders can be extended to relative number fields and orders.

Bug Fixes:

• RootOfUnity now returns a root when the order of the field is odd and the exponent given divides twice the order instead of giving an error when the exponent was not equal to twice the order.