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Package sage :: Package rings :: Module number_field :: Class NumberField |
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Gens
--+ |Ring
--+ |object
--+ | | |_uniqNumberField
--+ | NumberField
CyclotomicField
,
QuadraticField
Method Summary | |
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__init__(self,
poly,
variable,
check)
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Coerce x into this number field. | |
__cmp__(self,
other)
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__contains__(self,
x)
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__repr__(self)
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characteristic(self)
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WARNING: Assume GRH, etc. | |
class_number(self)
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List of all possible composite fields formed from self and other. | |
degree(self)
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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. | |
Ideal factorization of the principal ideal of the ring of integers generated by n. | |
gen(self,
n)
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Return a list of elements of this number field that are a basis for the full ring of integers. | |
is_field(self)
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name(self)
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ngens(self)
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order_table(self)
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PARI big number field corresponding to this field. | |
PARI number field corresponding to this field. | |
PARI polynomial corresponding to polynomial that defines this field. | |
polynomial(self)
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polynomial_ring(self)
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Returns the trace pairing on the elements of the list v. | |
units(self)
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Returns or sets the name of the generator of the number field. | |
zeta(self)
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Inherited from object | |
x.__delattr__('name') <==> del x.name | |
x.__getattribute__('name') <==> x.name | |
x.__hash__() <==> hash(x) | |
helper for pickle | |
helper for pickle | |
x.__setattr__('name', value) <==> x.name = value | |
x.__str__() <==> str(x) | |
Inherited from Ring | |
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. | |
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Inherited from Gens | |
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Instance Method Details |
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__call__(self,
x)
Coerce x into this number field.
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class_group(self)WARNING: Assume GRH, etc. !! TODO: Change to use bnf_certify, unless user requests not to. |
composite_fields(self, other)List of all possible composite fields formed from self and other. |
discriminant(self, v=None)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. INPUT: v (optional) -- list of element of this number field OUTPUT: Integer if v is omitted, and Rational otherwise. |
factor_integer(self, n)Ideal factorization of the principal ideal of the ring of integers generated by n. |
integral_basis(self)Return a list of elements of this number field that are a basis for the full ring of integers. EXAMPLES:>>> from sage.all import * >>> K = NumberField(x**5+10*x+1) >>> K.integral_basis() [1, a, a^2, a^3, a^4]Next we compute the ring of integers of a cubic field in which 2 is an "essential discriminant divisor", so the ring of integers is not generated by a single element. >>> K = NumberField(x**3 + x**2 - 2*x + 8) >>> K.integral_basis() [1, a, 1/2*a^2 + 1/2*a] |
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. |
trace_pairing(self, v)Returns the trace pairing on the elements of the list v. |
variable(self, new_var=None)Returns or sets the name of the generator of the number field. If the optional string argument new_var is given, then the name of the number field generator is set to new_var. If it is not given, this function returns the current name of the generator. INPUT: new_value -- optional str OUTPUT: str -- the name of the generator |
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