polynomial, [name=True], [check=None]) |
characteristic,
class_group,
class_number,
composite_fields,
degree,
disc,
discriminant,
factor_integer,
gen,
integral_basis,
is_field,
ngens,
order,
order_table,
pari_bnf,
pari_nf,
pari_polynomial,
polynomial,
polynomial_quotient_ring,
polynomial_ring,
trace_pairing,
units,
zeta
Further documentation:
) |
WARNING: Assume GRH, etc. !! TODO: Change to use bnf_certify, unless user requests not to.
other) |
List of all possible composite fields formed from self and other.
[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.
[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.
n) |
Ideal factorization of the principal ideal of the ring of integers generated by n.
) |
Return a list of elements of this number field that are a basis for the full ring of integers.
sage: x = PolynomialRing(QQ).gen() sage: K = NumberField(x**5+10*x+1, 'a') sage: 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.
sage: K = NumberField(x**3 + x**2 - 2*x + 8, 'a') sage: K.integral_basis() [1, a, 1/2*a^2 + 1/2*a]
) |
PARI big number field corresponding to this field.
) |
PARI number field corresponding to this field.
) |
PARI polynomial corresponding to polynomial that defines this field.
) |
Return the polynomial quotient ring isomorphic to this number field.
sage: R = PolynomialRing(RationalField(), 'x'); x = R.gen() sage: K = NumberField(x**3 + 2*x - 5, 'alpha') sage: K.polynomial_quotient_ring() Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^3 + 2*x - 5
v) |
Returns the trace pairing on the elements of the list v.
Instances of class NumberField_generic also have the following special methods:
__call__,
__cmp__,
__repr__,
_coerce_,
_latex_
Further documentation:
x) |
Coerce x into this number field.
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