4.6.1.1 CyclotomicField Objects

class CyclotomicField
Create a cyclotomic extension of the rational field.

The command CyclotomicField(n) creates the n-th cyclotomic field, got by adjoing an n-th root of unity to the rational field.

sage: CyclotomicField(3)
Cyclotomic Field of order 3 and degree 2
sage: CyclotomicField(18)
Cyclotomic Field of order 18 and degree 6
sage: z = CyclotomicField(6).gen(); z
zeta_6
sage: z**3
-1
sage: (1+z)**3
6*zeta_6 - 3
CyclotomicField( n)

Instances of class CyclotomicField have the following methods (in addition to inherited methods and special methods):

integral_basis

These methods are defined as follows:

integral_basis( )

Return a list of elements of this number field that are a basis for the full ring of integers.

Instances of class CyclotomicField also have the following special methods:

__call__( x)

The following example illustrates coercion from the cyclotomic field Q(zeta_42) to the cyclotomic field Q(zeta_6), in a case where such coercion is defined:

sage: k42 = CyclotomicField(42)
sage: k6 = CyclotomicField(6)
sage: a = k42.g0
sage: b = a**7
sage: b
zeta_42^7
sage: k6(b)
zeta_6
sage: b**2
zeta_42^7 - 1
sage: k6(b**2)
zeta_6 - 1

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