Ideals
(Integral) ideals in orders are always free $Z$-module of the same rank as the order, hence have a representation via a $Z$-basis. This can be made unique by normalising the corresponding matrix to be in reduced row echelon form (HNF).
For ideals in maximal orders $Z_K$, we also have a second presentation coming from the $Z_K$ module structure and the fact that $Z_K$ is a Dedekind ring: ideals can be generated by 2 elements, one of which can be any non-zero element in the ideal.
For efficiency, we will choose the 1st generator to be an integer.
Ideals here are of type AbsNumFieldOrderIdeal
, which is, similar to the elements above, also indexed by the type of the field and their elements: AbsNumFieldOrderIdeal{AbsSimpleNumField,AbsSimpleNumFieldElem}
for ideals in simple absolute fields.
Different to elements, the parent
of an ideal is the set of all ideals in the ring, of type AbsNumFieldOrderIdealSet
.
Creation
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, a::Integer) -> AbsNumFieldOrderIdeal
Returns the ideal of $\mathcal O$ which is generated by $a$.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, M::ZZMatrix; check::Bool = false, M_in_hnf::Bool = false) -> AbsNumFieldOrderIdeal
Creates the ideal of $\mathcal O$ with basis matrix $M$. If check
is set, then it is checked whether $M$ defines an ideal (expensive). If M_in_hnf
is set, then it is assumed that $M$ is already in lower left HNF.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the principal ideal $(x)$ of $\mathcal O$.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the ideal $(x, y)$ of $\mathcal O$.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the ideal $(x, y)$ of $\mathcal O$.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, a::Integer) -> AbsNumFieldOrderIdeal
Returns the ideal of $\mathcal O$ which is generated by $a$.
ideal
— Methodideal(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the principal ideal $(x)$ of $\mathcal O$.
*
— Method*(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
*(x::AbsNumFieldOrderElem, O::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
Returns the principal ideal $(x)$ of $\mathcal O$.
factor
— Methodfactor(a::T) where T <: RingElement -> Fac{T}
Return a factorization of $a$ into irreducible elements, as a Fac{T}
. The irreducible elements in the factorization are pairwise coprime.
factor(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
Computes the prime ideal factorization $A$ as a dictionary, the keys being the prime ideal divisors: If lp = factor_dict(A)
, then keys(lp)
are the prime ideal divisors of $A$ and lp[P]
is the $P$-adic valuation of $A$ for all $P$ in keys(lp)
.
factor
— Methodfactor(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::AbsSimpleNumFieldElem) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
Factors the principal ideal generated by $a$.
coprime_base
— Methodcoprime_base(A::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}
coprime_base(A::Vector{AbsSimpleNumFieldOrderElem}) -> Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}
A coprime base for the (principal) ideals in $A$, i.e. the returned array generated multiplicatively the same ideals as the input and are pairwise coprime.
Arithmetic
All the usual operations are supported:
==
,+
,*
divexact
,divides
lcm
,gcd
in
intersect
— Methodintersect(x::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, y::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Returns $x \cap y$.
colon
— Methodcolon(a::AbsNumFieldOrderIdeal, b::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal
The ideal $(a:b) = \{x \in K | xb \subseteq a\} = \hom(b, a)$ where $K$ is the number field.
in
— Methodin(x::NumFieldOrderElem, y::NumFieldOrderIdeal)
in(x::NumFieldElem, y::NumFieldOrderIdeal)
in(x::ZZRingElem, y::NumFieldOrderIdeal)
Returns whether $x$ is contained in $y$.
is_power
— Methodis_power(A::AbsNumFieldOrderIdeal, n::Int) -> Bool, AbsNumFieldOrderIdeal
is_power(A::AbsSimpleNumFieldOrderFractionalIdeal, n::Int) -> Bool, AbsSimpleNumFieldOrderFractionalIdeal
Computes, if possible, an ideal $B$ s.th. $B^n==A$ holds. In this case, true
and $B$ are returned.
is_power
— Methodis_power(I::AbsNumFieldOrderIdeal) -> Int, AbsNumFieldOrderIdeal
is_power(a::AbsSimpleNumFieldOrderFractionalIdeal) -> Int, AbsSimpleNumFieldOrderFractionalIdeal
Writes $a = r^e$ with $e$ maximal. Note: $1 = 1^0$.
is_invertible
— Methodis_invertible(A::AbsNumFieldOrderIdeal) -> Bool, AbsSimpleNumFieldOrderFractionalIdeal
Returns true
and an inverse of $A$ or false
and an ideal $B$ such that $A*B \subsetneq order(A)$, if $A$ is not invertible.
isone
— Methodisone(A::AbsNumFieldOrderIdeal) -> Bool
is_unit(A::AbsNumFieldOrderIdeal) -> Bool
Tests if $A$ is the trivial ideal generated by $1$.
Class Group
The group of invertable ideals in any order forms a group and the principal ideals a subgroup. The finite quotient is called class group for maximal orders and Picard group or ring class group in general.
class_group
— Methodclass_group(O::AbsSimpleNumFieldOrder; bound = -1,
redo = false,
GRH = true) -> FinGenAbGroup, Map
Returns a group $A$ and a map $f$ from $A$ to the set of ideals of $O$. The inverse of the map is the projection onto the group of ideals modulo the group of principal ideals.
By default, the correctness is guarenteed only assuming the Generalized Riemann Hypothesis (GRH).
Keyword arguments:
redo
: Trigger a recomputation, thus avoiding the cache.bound
: When specified, this is used for the bound for the factor base.GRH
: Iffalse
, the correctness of the result does not depend on GRH.
narrow_class_group
— Methodnarrow_class_group(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map
Computes the narrow (or strict) class group of $O$, ie. the group of invertable ideals modulo principal ideals generated by elements that are positive at all real places.
picard_group
— Methodpicard_group(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, MapClassGrp
Returns the Picard group of $O$ and a map from the group in the set of (invertible) ideals of $O$.
ring_class_group
— Methodring_class_group(O::AbsNumFieldOrder)
The ring class group (Picard group) of $O$.
julia> k, a = wildanger_field(3, 13);
julia> zk = maximal_order(k);
julia> c, mc = class_group(zk)
(Z/9, ClassGroup map of Set of ideals of Maximal order of Number field of degree 3 over QQ with basis AbsSimpleNumFieldElem[1, _$, 1//2*_$^2 + 1//2])
julia> lp = prime_ideals_up_to(zk, 20);
julia> [ mc \ I for I = lp]
10-element Vector{FinGenAbGroupElem}: [7] [1] [1] [8] [3] [5] [4] [7] [0] [5]
julia> mc(c[1])
<2, 7//2*_$^2 + 2*_$ + 7//2> Norm: 2 Minimum: 2 two normal wrt: 2
julia> order(c[1])
9
julia> mc(c[1])^Int(order(c[1]))
<32, 51324932762088712549357//2*_$^2 - 25469765820220196858689*_$ + 55567239093774188249155//2> Norm: 512 Minimum: 32 two normal wrt: 2
julia> mc \ ans
Abelian group element [0]
The class group, or more precisely the information used to compute it also allows for principal ideal testing and related tasks. In general, due to the size of the objects, the fac_elem
versions are more efficient.
is_principal
— Methodis_principal(A::AbsSimpleNumFieldOrderIdeal) -> Bool
is_principal(A::AbsSimpleNumFieldOrderFractionalIdeal) -> Bool
Tests if $A$ is principal.
is_principal_with_data
— Methodis_principal_with_data(A::AbsSimpleNumFieldOrderIdeal) -> Bool, AbsSimpleNumFieldOrderElem
is_principal_with_data(A::AbsSimpleNumFieldOrderFractionalIdeal) -> Bool, AbsSimpleNumFieldElem
Tests if $A$ is principal and returns $(\mathtt{true}, \alpha)$ if $A = \langle \alpha\rangle$ or $(\mathtt{false}, 1)$ otherwise.
is_principal_fac_elem
— Methodis_principal_fac_elem(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool, FacElem{AbsSimpleNumFieldElem, number_field}
Tests if $A$ is principal and returns $(\mathtt{true}, \alpha)$ if $A = \langle \alpha\rangle$ or $(\mathtt{false}, 1)$ otherwise. The generator will be in factored form.
power_class
— Methodpower_class(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, e::ZZRingElem) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Computes a (small) ideal in the same class as $A^e$.
power_product_class
— Methodpower_product_class(A::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}, e::Vector{ZZRingElem}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Computes a (small) ideal in the same class as $\prod A_i^{e_i}$.
power_reduce
— Methodpower_reduce(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, e::ZZRingElem) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
Computes $B$ and $\alpha$ in factored form, such that $\alpha B = A^e$ $B$ has small norm.
class_group_ideal_relation
— Methodclass_group_ideal_relation(I::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, c::ClassGrpCtx) -> AbsSimpleNumFieldElem, SRow{ZZRingElem}
Finds a number field element $\alpha$ such that $\alpha I$ factors over the factor base in $c$.
factor_base_bound_grh
— Methodfactor_base_bound_grh(O::AbsSimpleNumFieldOrder) -> Int
Returns an integer $B$, such that under GRH the ideal class group of $\mathcal O$ is generated by the prime ideals of norm bounded by $B$.
factor_base_bound_bach
— Methodfactor_base_bound_bach(O::AbsSimpleNumFieldOrder) -> Int
Use the theorem of Bach to find $B$ such that under GRH the ideal class group of $\mathcal O$ is generated by the prime ideals of norm bounded by $B$.
prime_ideals_up_to
— Functionprime_ideals_up_to(O::AbsSimpleNumFieldOrder,
B::Int;
degree_limit::Int = 0, index_divisors::Bool = true) -> Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}
Computes the prime ideals $\mathcal O$ with norm up to $B$.
If degree_limit
is a nonzero integer $k$, then prime ideals $\mathfrak p$ with $\deg(\mathfrak p) > k$ will be discarded. If 'index_divisors' is set to false, only primes not dividing the index of the order will be computed.
prime_ideals_up_to(O::AbsSimpleNumFieldOrder,
B::Int;
complete::Bool = false,
degree_limit::Int = 0,
F::Function,
bad::ZZRingElem)
Computes the prime ideals $\mathcal O$ with norm up to $B$.
If degree_limit
is a nonzero integer $k$, then prime ideals $\mathfrak p$ with $\deg(\mathfrak p) > k$ will be discarded.
The function $F$ must be a function on prime numbers not dividing bad
such that $F(p) = \deg(\mathfrak p)$ for all prime ideals $\mathfrak p$ lying above $p$.
julia> I = mc(c[1])
<2, 7//2*_$^2 + 2*_$ + 7//2> Norm: 2 Minimum: 2 two normal wrt: 2
julia> is_principal(I)
false
julia> I = I^Int(order(c[1]))
<32, 51324932762088712549357//2*_$^2 - 25469765820220196858689*_$ + 55567239093774188249155//2> Norm: 512 Minimum: 32 two normal wrt: 2
julia> is_principal(I)
true
julia> is_principal_fac_elem(I)
(true, 5^1*(_$ - 3)^-1*(23//2*_$^2 - 126*_$ - 221//2)^4*(_$^2 + _$ + 2)^-1*(_$ + 5)^2*(_$^2 + 1)^2*3^-2*(-6*_$^2 + 67*_$ - 130)^-4*11^-1*(_$ + 1)^-1*(_$ + 2)^2*1^-1)
The computation of $S$-units is also tied to the class group:
torsion_units
— Methodtorsion_units(O::AbsSimpleNumFieldOrder) -> Vector{AbsSimpleNumFieldOrderElem}
Given an order $O$, compute the torsion units of $O$.
torsion_unit_group
— Methodtorsion_unit_group(O::AbsSimpleNumFieldOrder) -> GrpAb, Map
Given an order $\mathcal O$, returns the torsion units as an abelian group $G$ together with a map $G \to \mathcal O^\times$.
torsion_units_generator
— Methodtorsion_units_generator(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem
Given an order $O$, compute a generator of the torsion units of $O$.
torsion_units_gen_order
— Methodtorsion_units_gen_order(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem
Given an order $O$, compute a generator of the torsion units of $O$ as well as its order.
unit_group
— Methodunit_group(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map
Returns a group $U$ and an isomorphism map $f \colon U \to \mathcal O^\times$. A set of fundamental units of $\mathcal O$ can be obtained via [ f(U[1+i]) for i in 1:unit_group_rank(O) ]
. f(U[1])
will give a generator for the torsion subgroup.
unit_group_fac_elem
— Methodunit_group_fac_elem(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map
Returns a group $U$ and an isomorphism map $f \colon U \to \mathcal O^\times$. A set of fundamental units of $\mathcal O$ can be obtained via [ f(U[1+i]) for i in 1:unit_group_rank(O) ]
. f(U[1])
will give a generator for the torsion subgroup. All elements will be returned in factored form.
sunit_group
— Methodsunit_group(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map
For an array $I$ of (coprime prime) ideals, find the $S$-unit group defined by $I$, ie. the group of non-zero field elements which are only divisible by ideals in $I$.
sunit_group_fac_elem
— Methodsunit_group_fac_elem(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map
For an array $I$ of (coprime prime) ideals, find the $S$-unit group defined by $I$, ie. the group of non-zero field elements which are only divisible by ideals in $I$. The map will return elements in factored form.
sunit_mod_units_group_fac_elem
— Methodsunit_mod_units_group_fac_elem(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map
For an array $I$ of (coprime prime) ideals, find the $S$-unit group defined by $I$, ie. the group of non-zero field elements which are only divisible by ideals in $I$ modulo the units of the field. The map will return elements in factored form.
julia> u, mu = unit_group(zk)
(Z/2 x Z, UnitGroup map of Maximal order of Number field of degree 3 over QQ with basis AbsSimpleNumFieldElem[1, _$, 1//2*_$^2 + 1//2] )
julia> mu(u[2])
_$^2 - _$ + 1
julia> u, mu = unit_group_fac_elem(zk)
(Z/2 x Z, UnitGroup map of Factored elements over Number field of degree 3 over QQ )
julia> mu(u[2])
(_$ - 4)^1*(-22*_$^2 + 314*_$ - 428)^-1*(_$^2 + 1)^1*(-_$ + 5)^-1*(-46*_$^2 + 576*_$ - 28)^-1*(-35//2*_$^2 + 214*_$ - 599//2)^1*(1//2*_$^2 - 5*_$ - 3//2)^-1*(_$ - 3)^1*(-33//2*_$^2 + 214*_$ - 57//2)^1
julia> evaluate(ans)
_$^2 - _$ + 1
julia> lp = factor(6*zk)
Dict{AbsSimpleNumFieldOrderIdeal, Int64} with 4 entries: <3, _$ + 5> => 1 <3, _$^2 + 1> => 1 <2, 7//2*_$^2 + 2*_$ + 7//2> => 2 <2, 3//2*_$^2 + _$ + 3//2> => 1
julia> s, ms = Hecke.sunit_group(collect(keys(lp)))
(Z/2 x Z^(5), SUnits map of Number field of degree 3 over QQ for AbsSimpleNumFieldOrderIdeal[<3, _$ + 5> Norm: 3 Minimum: 3 basis_matrix [3 0 0; 2 1 0; 2 0 1] two normal wrt: 3, <3, _$^2 + 1> Norm: 9 Minimum: 3 basis_matrix [3 0 0; 0 3 0; 0 0 1] two normal wrt: 3, <2, 7//2*_$^2 + 2*_$ + 7//2> Norm: 2 Minimum: 2 basis_matrix [2 0 0; 1 1 0; 0 0 1] two normal wrt: 2, <2, 3//2*_$^2 + _$ + 3//2> Norm: 2 Minimum: 2 basis_matrix [2 0 0; 1 1 0; 1 0 1] two normal wrt: 2] )
julia> ms(s[4])
-1//2*_$^2 + 6*_$ + 5//2
julia> norm(ans)
144
julia> factor(numerator(ans))
1 * 2^4 * 3^2
Miscaellenous
order
— Methodorder(::Type{T} = BigInt, G::Group) where T
Return the order of $G$ as an instance of T
. If $G$ is of infinite order, an InfiniteOrderError
exception will be thrown. Use is_finite(G)
to avoid this kind of exception. If the order does not fit into type T
, an InexactError
exception will be thrown.
order(::Type{T} = BigInt, g::GroupElem) where T
Return the order of $g$ as an instance of T
. If $g$ is of infinite order, an InfiniteOrderError
exception will be thrown. Use is_finite_order(G)
to avoid this kind of exception. If the order does not fit into type T
, an InexactError
exception will be thrown.
order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder
Returns the order of $I$.
order(::Type{T} = ZZRingElem, c::CycleType) where T <: IntegerUnion
Return the order of the permutations with cycle structure c
.
Examples
julia> g = symmetric_group(3);
julia> all(x -> order(cycle_structure(x)) == order(x), gens(g))
true
order(::Type{T}, W::WeylGroup) where {T} -> T
Returns the order of W
.
This function is part of the experimental code in Oscar. Please read here for more details.
order
— Methodorder(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder
The order that was used to define the ideal $a$.
order
— Methodorder(::Type{T} = BigInt, G::Group) where T
Return the order of $G$ as an instance of T
. If $G$ is of infinite order, an InfiniteOrderError
exception will be thrown. Use is_finite(G)
to avoid this kind of exception. If the order does not fit into type T
, an InexactError
exception will be thrown.
order(::Type{T} = BigInt, g::GroupElem) where T
Return the order of $g$ as an instance of T
. If $g$ is of infinite order, an InfiniteOrderError
exception will be thrown. Use is_finite_order(G)
to avoid this kind of exception. If the order does not fit into type T
, an InexactError
exception will be thrown.
order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder
Returns the order of $I$.
order(::Type{T} = ZZRingElem, c::CycleType) where T <: IntegerUnion
Return the order of the permutations with cycle structure c
.
Examples
julia> g = symmetric_group(3);
julia> all(x -> order(cycle_structure(x)) == order(x), gens(g))
true
order(::Type{T}, W::WeylGroup) where {T} -> T
Returns the order of W
.
This function is part of the experimental code in Oscar. Please read here for more details.
order
— Methodorder(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrder
Returns the order of $a$.
nf
— Methodnf(x::NumFieldOrderIdeal) -> AbsSimpleNumField
Returns the number field, of which $x$ is an integral ideal.
basis
— Methodbasis(A::AbsNumFieldOrderIdeal) -> Vector{AbsSimpleNumFieldOrderElem}
Returns the basis of $A$.
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
Returns the $\mathbf Z$-basis of $I$.
lll_basis
— Methodlll_basis(I::NumFieldOrderIdeal) -> Vector{NumFieldElem}
A basis for $I$ that is reduced using the LLL algorithm for the Minkowski metric.
basis_matrix
— Methodbasis_matrix(A::AbsNumFieldOrderIdeal) -> ZZMatrix
Returns the basis matrix of $A$.
basis_mat_inv
— Methodbasis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
Return the inverse of the basis matrix of $A$.
has_princ_gen_special
— Methodhas_princ_gen_special(A::AbsNumFieldOrderIdeal) -> Bool
Returns whether $A$ knows if it is generated by a rational integer.
principal_generator
— Methodprincipal_generator(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem
For a principal ideal $A$, find a generator.
principal_generator_fac_elem
— Methodprincipal_generator_fac_elem(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> FacElem{AbsSimpleNumFieldElem, number_field}
For a principal ideal $A$, find a generator in factored form.
minimum
— Methodminimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem
Returns the smallest non-negative element in $A \cap \mathbf Z$.
minimum(A::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
minimum(A::RelNumFieldOrderIdeal) -> RelNumFieldOrderIdeal
Returns the ideal $A \cap O$ where $O$ is the maximal order of the coefficient ideals of $A$.
minimum
— Method minimum(A::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
minimum(A::RelNumFieldOrderIdeal) -> RelNumFieldOrderIdeal
Returns the ideal $A \cap O$ where $O$ is the maximal order of the coefficient ideals of $A$.
minimum
— Methodminimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem
Returns the smallest non-negative element in $A \cap \mathbf Z$.
has_minimum
— Methodhas_minimum(A::AbsNumFieldOrderIdeal) -> Bool
Returns whether $A$ knows its minimum.
norm
— Methodnorm(A::AbsNumFieldOrderIdeal) -> ZZRingElem
Returns the norm of $A$, that is, the cardinality of $\mathcal O/A$, where $\mathcal O$ is the order of $A$.
norm(a::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Returns the norm of $a$.
norm(a::RelNumFieldOrderFractionalIdeal{T, S}) -> S
Returns the norm of $a$.
norm(a::AlgAssAbsOrdIdl, O::AlgAssAbsOrd; copy::Bool = true) -> QQFieldElem
Returns the norm of $a$ considered as an (possibly fractional) ideal of $O$.
norm(a::AlgAssRelOrdIdl{S, T, U}, O::AlgAssRelOrd{S, T, U}; copy::Bool = true)
where { S, T, U } -> T
Returns the norm of $a$ considered as an (possibly fractional) ideal of $O$.
has_norm
— Methodhas_norm(A::AbsNumFieldOrderIdeal) -> Bool
Returns whether $A$ knows its norm.
idempotents
— Methodidempotents(x::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, y::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem, AbsSimpleNumFieldOrderElem
Returns a tuple (e, f)
consisting of elements e in x
, f in y
such that 1 = e + f
.
If the ideals are not coprime, an error is raised.
is_prime
— Methodis_prime(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool
Returns whether $A$ is a prime ideal.
is_prime_known
— Methodis_prime_known(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool
Returns whether $A$ knows if it is prime.
is_ramified
— Methodis_ramified(O::AbsSimpleNumFieldOrder, p::Int) -> Bool
Returns whether the integer $p$ is ramified in $\mathcal O$. It is assumed that $p$ is prime.
ramification_index
— Methodramification_index(P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
The ramification index of the prime-ideal $P$.
degree
— Methoddegree(P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
The inertia degree of the prime-ideal $P$.
valuation
— Methodvaluation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the $\mathfrak p$-adic valuation of $a$, that is, the largest $i$ such that $a$ is contained in $\mathfrak p^i$.
valuation
— Methodvaluation(a::AbsSimpleNumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
valuation(a::AbsSimpleNumFieldOrderElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
valuation(a::ZZRingElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the $\mathfrak p$-adic valuation of $a$, that is, the largest $i$ such that $a$ is contained in $\mathfrak p^i$.
valuation
— Methodvaluation(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the $\mathfrak p$-adic valuation of $A$, that is, the largest $i$ such that $A$ is contained in $\mathfrak p^i$.
valuation
— Methodvaluation(a::Integer, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the $\mathfrak p$-adic valuation of $a$, that is, the largest $i$ such that $a$ is contained in $\mathfrak p^i$.
valuation
— Methodvaluation(a::AbsSimpleNumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
valuation(a::AbsSimpleNumFieldOrderElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
valuation(a::ZZRingElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the $\mathfrak p$-adic valuation of $a$, that is, the largest $i$ such that $a$ is contained in $\mathfrak p^i$.
valuation
— Methodvaluation(A::AbsNumFieldOrderFractionalIdeal, p::AbsNumFieldOrderIdeal)
The valuation of $A$ at $p$.
idempotents
— Methodidempotents(x::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, y::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem, AbsSimpleNumFieldOrderElem
Returns a tuple (e, f)
consisting of elements e in x
, f in y
such that 1 = e + f
.
If the ideals are not coprime, an error is raised.
Quotient Rings
quo
— Methodquo(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderQuoRing, Map
quo(O::AlgAssAbsOrd, I::AlgAssAbsOrdIdl) -> AbsOrdQuoRing, Map
The quotient ring $O/I$ as a ring together with the section $M: O/I \to O$. The pointwise inverse of $M$ is the canonical projection $O\to O/I$.
residue_ring
— Methodresidue_ring(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderQuoRing
residue_ring(O::AlgAssAbsOrd, I::AlgAssAbsOrdIdl) -> AbsOrdQuoRing
The quotient ring $O$ modulo $I$ as a new ring.
residue_field
— Methodresidue_field(O::AbsSimpleNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, check::Bool = true) -> Field, Map
Returns the residue field of the prime ideal $P$ together with the projection map. If check
is true, the ideal is checked for being prime.
mod
— Methodmod(x::AbsSimpleNumFieldOrderElem, I::AbsNumFieldOrderIdeal)
Returns the unique element $y$ of the ambient order of $x$ with $x \equiv y \bmod I$ and the following property: If $a_1,\dotsc,a_d \in \mathbf{Z}_{\geq 1}$ are the diagonal entries of the unique HNF basis matrix of $I$ and $(b_1,\dotsc,b_d)$ is the coefficient vector of $y$, then $0 \leq b_i < a_i$ for $1 \leq i \leq d$.
crt
— Methodcrt(r1::AbsSimpleNumFieldOrderElem, i1::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, r2::AbsSimpleNumFieldOrderElem, i2::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem
Find $x$ such that $x \equiv r_1 \bmod i_1$ and $x \equiv r_2 \bmod i_2$ using idempotents
.
euler_phi
— Methodeuler_phi(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
The ideal version of the totient function returns the size of the unit group of the residue ring modulo the ideal.
multiplicative_group
— Methodmultiplicative_group(Q::AbsSimpleNumFieldOrderQuoRing) -> FinGenAbGroup, Map{FinGenAbGroup, AbsSimpleNumFieldOrderQuoRing}
unit_group(Q::AbsSimpleNumFieldOrderQuoRing) -> FinGenAbGroup, Map{FinGenAbGroup, AbsSimpleNumFieldOrderQuoRing}
Returns the unit group of $Q$ as an abstract group $A$ and an isomorphism map $f \colon A \to Q^\times$.
multiplicative_group_generators
— Methodmultiplicative_group_generators(Q::AbsSimpleNumFieldOrderQuoRing) -> Vector{AbsSimpleNumFieldOrderQuoRingElem}
Return a set of generators for $Q^\times$.