Cyclotomic Fields (like the Quadratic Fields) are a subtype of the Number
Fields (FldNum). They have some extra functionality which is described
below and use some more efficient implementations. Orders of cyclotomic fields
form the category RngCyc and the fields themselves FldCyc.
Functions for cyclotomic fields and orders which work generally for number
fields, their orders and elements are listed in Chapter NUMBER FIELDS.
There are two different representations of cyclotomic fields available:
- *
- The "dense" representation: the field is conceptually represented as
Q(x)/f(x) where f is a cyclotomic polynomial, i.e., the minimal
polynomial of a primitive root of unity.
- *
- The "sparse" representation: Let n=∏piri be the factorisation
of n into prime powers and ni := piri.
Then Q(ζn) = Q(ζn1, ..., ζnr) and the field
is represented as
Q(x1, ..., xr)/< fn1(x1), ..., fnr(xr) >.
As with the number fields, the non-simple representation, the issues are the
same: the "sparse" representation allows for much larger fields -- as long
as the elements used have only few coefficients. The "dense" representation
on the other hand has the asymptotically-fastest arithmetic.
V2.28, 13 July 2023