CurrentModule = Hecke
DocTestSetup = quote
using Hecke
end
General number fields can be created using the function number_field.
To create a simple number field given by a defining
polynomial or a non-simple number field given by defining polynomials, the
following functions can be used.
number_field(::DocuDummy)
number_field(::DocuDummy2)
!!! tip
Many of the constructors have arguments of type Symbol or
AbstractString. If used, they define the appearance in printing, and
printing only. The named parameter check can be true or false, the
default being true. This parameter controls whether the polynomials
defining the number field are tested for irreducibility or not. Given that
this can be potentially very time consuming if the degree if large, one can
disable this test. Note however, that the behaviour of Hecke is undefined
if a reducible polynomial is used to define a field.
The named boolean parameter `cached` can be used to disable caching. Two
number fields defined using the same polynomial from the identical
polynomial ring and the same (identical) symbol/string will be identical if
`cached == true` and different if `cached == false`.
For frequently used number fields like quadratic fields, cyclotomic fields or radical extensions, the following functions are provided:
cyclotomic_field(n::Int)
quadratic_field(d::ZZRingElem)
wildanger_field(n::Int, B::ZZRingElem)
radical_extension(n::Int, a::NumFieldElem)
rationals_as_number_field()
basis(::SimpleNumField)
basis(::NonSimpleNumField)
absolute_basis(::NumField)
defining_polynomial(::SimpleNumField)
defining_polynomials(::NonSimpleNumField)
absolute_primitive_element(::NumField)
component(::NonSimpleNumField, ::Int)
base_field(::NumField)
degree(::NumField)
absolute_degree(::NumField)
signature(::NumField)
unit_group_rank(::NumField)
class_number(::AbsSimpleNumField)
relative_class_number(::AbsSimpleNumField)
regulator(::AbsSimpleNumField)
discriminant(::SimpleNumField)
absolute_discriminant(::SimpleNumField)
is_simple(::NumField)
is_absolute(::NumField)
is_totally_real(::NumField)
is_totally_complex(::NumField)
is_cm_field(::NumField)
is_kummer_extension(::SimpleNumField)
is_radical_extension(::SimpleNumField)
is_linearly_disjoint(::SimpleNumField, ::SimpleNumField)
is_weakly_ramified(::AbsSimpleNumField, ::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
is_tamely_ramified(::AbsSimpleNumField)
is_tamely_ramified(::AbsSimpleNumField, p::Int)
is_abelian(::NumField)
is_subfield(::SimpleNumField, ::SimpleNumField)
subfields(::SimpleNumField)
principal_subfields(::SimpleNumField)
compositum(::AbsSimpleNumField, ::AbsSimpleNumField)
embedding(::NumField, ::NumField)
normal_closure(::AbsSimpleNumField)
relative_simple_extension(::NumField, ::NumField)
is_subfield_normal(::AbsSimpleNumField, ::AbsSimpleNumField)
simplify(::AbsSimpleNumField)
absolute_simple_field(K::NumField)
simple_extension(::NonSimpleNumField)
simplified_simple_extension(::NonSimpleNumField)
is_isomorphic(::SimpleNumField, ::SimpleNumField)
is_isomorphic_with_map(::SimpleNumField, ::SimpleNumField)
is_involution(::NumFieldHom{AbsSimpleNumField, AbsSimpleNumField})
fixed_field(::NumFieldHom)
automorphism_list(::NumField)
automorphism_group(::AbsSimpleNumField)
complex_conjugation(::AbsSimpleNumField)
normal_basis(::NumField)
decomposition_group(::AbsSimpleNumField, ::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ::Map)
ramification_group(::AbsSimpleNumField, ::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ::Int, ::Map)
inertia_subgroup(::AbsSimpleNumField, ::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ::Map)
infinite_places(K::NumField)
real_places(K::AbsSimpleNumField)
complex_places(K::AbsSimpleNumField)
isreal(::Plc)
is_complex(::Plc)
norm_equation(::AbsSimpleNumField, ::Any)
lorenz_module(::AbsSimpleNumField, ::Int)
kummer_failure(::AbsSimpleNumFieldElem, ::Int, ::Int)
is_defining_polynomial_nice(::AbsSimpleNumField)