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

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


Create a quadratic extension of the rational field.

The command QuadraticExtension(a) creates the field Q(sqrt(a)).

EXAMPLES:
>>> QuadraticField(3)
Number Field with defining polynomial x^2 - 3

>>> QuadraticField(-4)
Number Field with defining polynomial x^2 + 4

Method Summary
  __init__(self, f, variable, check)
  class_number(self, proof)
Return the size of the class group of self.
  disc(self, v)
    Inherited from NumberField
  __call__(self, x)
Coerce x into this number field.
  __cmp__(self, other)
  __contains__(self, x)
  __repr__(self)
  characteristic(self)
  class_group(self)
WARNING: Assume GRH, etc.
  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)
  integral_basis(self)
Return a list of elements of this number field that are a basis for the full ring of integers.
  is_field(self)
  name(self)
  ngens(self)
  order_table(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.
  zeta(self)
    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

class_number(self, proof=True)

Return the size of the class group of self.

If proof = False (not the default) and the discriminant of the field is negative, then the following warning from the PARI manual applies: IMPORTANT WARNING: For D<0, this function may give incorrect results when the class group has a low exponent (has many cyclic factors), because implementing Shank's method in full generality slows it down immensely.
Overrides:
sage.rings.number_field.NumberField.class_number

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