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Package sage :: Package rings :: Module number_field :: Class CyclotomicField |
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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 | |
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__init__(self,
n)
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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)
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Return a list of elements of this number field that are a basis for the full ring of integers. | |
order(self)
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order_table(self)
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zeta(self)
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Inherited from NumberField | |
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WARNING: Assume GRH, etc. | |
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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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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. | |
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)
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:
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integral_basis(self)Return a list of elements of this number field that are a basis for the full ring of integers. |
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