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Package sage :: Package rings :: Module number_field :: Class QuadraticField |
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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 | |
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__init__(self,
f,
variable,
check)
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Return the size of the class group of self. | |
disc(self,
v)
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Inherited from NumberField | |
Coerce x into this number field. | |
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WARNING: Assume GRH, etc. | |
List of all possible composite fields formed from self and other. | |
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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. | |
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Return a list of elements of this number field that are a basis for the full ring of integers. | |
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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. | |
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Returns the trace pairing on the elements of the list v. | |
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Returns or sets the name of the generator of the number field. | |
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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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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. |
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