Package sage :: Package rings :: Module number_field :: Class CyclotomicField
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Type CyclotomicField

            Gens --+    
                   |    
            Ring --+    
                   |    
      object --+   |    
               |   |    
_uniqNumberField --+    
                   |    
         NumberField --+
                       |
                      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.

EXAMPLES:
>>> CyclotomicField(3)
Cyclotomic Field of order 3 and degree 2

>>> CyclotomicField(18)
Cyclotomic Field of order 18 and degree 6

>>> z = CyclotomicField(6).gen(); z
zeta_6

>>> z**3
-1

>>> (1+z)**3
6*zeta_6 - 3

Method Summary
  __init__(self, n)
  __call__(self, x)
EXAMPLES: 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:
  __repr__(self)
  integral_basis(self)
Return a list of elements of this number field that are a basis for the full ring of integers.
  order(self)
  order_table(self)
  zeta(self)
    Inherited from NumberField
  __cmp__(self, other)
  __contains__(self, x)
  characteristic(self)
  class_group(self)
WARNING: Assume GRH, etc.
  class_number(self)
  composite_fields(self, other)
List of all possible composite fields formed from self and other.
  degree(self)
  discriminant(self, v)
Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.
  factor_integer(self, n)
Ideal factorization of the principal ideal of the ring of integers generated by n.
  gen(self, n)
  is_field(self)
  name(self)
  ngens(self)
  pari_bnf(self)
PARI big number field corresponding to this field.
  pari_nf(self)
PARI number field corresponding to this field.
  pari_polynomial(self)
PARI polynomial corresponding to polynomial that defines this field.
  polynomial(self)
  polynomial_ring(self)
  trace_pairing(self, v)
Returns the trace pairing on the elements of the list v.
  units(self)
  variable(self, new_var)
Returns or sets the name of the generator of the number field.
    Inherited from object
  __delattr__(...)
x.__delattr__('name') <==> del x.name
  __getattribute__(...)
x.__getattribute__('name') <==> x.name
  __hash__(x)
x.__hash__() <==> hash(x)
  __reduce__(...)
helper for pickle
  __reduce_ex__(...)
helper for pickle
  __setattr__(...)
x.__setattr__('name', value) <==> x.name = value
  __str__(x)
x.__str__() <==> str(x)
    Inherited from Ring
  is_atomic_repr(self)
True if the elements have atomic string representations, in the sense that they print if they print at s, then -s means the negative of s.
  type(self)
    Inherited from Gens
  __getattr__(self, attrname)
  __getitem__(self, n)
  __getslice__(self, n, m)
  gens(self)
  list(self)

Instance Method Details

__call__(self, x)
(Call operator)

EXAMPLES: 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:
>>> k42 = CyclotomicField(42)
>>> k6 = CyclotomicField(6)
>>> a = k42.g0
>>> b = a**7
>>> b
zeta_42^7

>>> k6(b)
zeta_6

>>> b**2
zeta_42^7 - 1

>>> k6(b**2)
zeta_6 - 1
Overrides:
sage.rings.number_field.NumberField.__call__

integral_basis(self)

Return a list of elements of this number field that are a basis for the full ring of integers.
Overrides:
sage.rings.number_field.NumberField.integral_basis

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