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 parentof an ideal is the set of all ideals in the ring, of type AbsNumFieldOrderIdealSet.

Creation

ideal — Method

ideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, a::Integer) -> AbsNumFieldOrderIdeal

Returns the ideal of $\mathcal O$ which is generated by $a$.

ideal — Method

ideal(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 — Method

ideal(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal

Creates the principal ideal $(x)$ of $\mathcal O$.

ideal — Method

ideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal

Creates the ideal $(x, y)$ of $\mathcal O$.

ideal — Method

ideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal

Creates the ideal $(x, y)$ of $\mathcal O$.

ideal — Method

ideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
ideal(O::AbsSimpleNumFieldOrder, a::Integer) -> AbsNumFieldOrderIdeal

Returns the ideal of $\mathcal O$ which is generated by $a$.

ideal — Method

ideal(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 — Method

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 — Method

factor(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::AbsSimpleNumFieldElem) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}

Factors the principal ideal generated by $a$.

coprime_base — Method

coprime_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 — Method

intersect(x::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, y::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

Returns $x \cap y$.

colon — Method

colon(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 — Method

in(x::NumFieldOrderElem, y::NumFieldOrderIdeal)
in(x::NumFieldElem, y::NumFieldOrderIdeal)
in(x::ZZRingElem, y::NumFieldOrderIdeal)

Returns whether $x$ is contained in $y$.

is_power — Method

is_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 — Method

is_power(I::AbsNumFieldOrderIdeal) -> Int, AbsNumFieldOrderIdeal
is_power(a::AbsSimpleNumFieldOrderFractionalIdeal) -> Int, AbsSimpleNumFieldOrderFractionalIdeal

Writes $a = r^e$ with $e$ maximal. Note: $1 = 1^0$.

is_invertible — Method

is_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 — Method

isone(A::AbsNumFieldOrderIdeal) -> Bool
is_unit(A::AbsNumFieldOrderIdeal) -> Bool

Tests if $A$ is the trivial ideal generated by $1$.

Class Group

The group of invertible 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 — Method

class_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: If false, the correctness of the result does not depend on GRH.

narrow_class_group — Method

narrow_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 — Method

picard_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 — Method

ring_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, Class group map of set of ideals of zk)

julia> lp = prime_ideals_up_to(zk, 20);

julia> [ mc \ I for I = lp]
10-element Vector{FinGenAbGroupElem}:
 [1]
 [4]
 [4]
 [5]
 [3]
 [2]
 [7]
 [1]
 [0]
 [2]

julia> mc(c[1])
<2, 3//2*_$^2 + 2*_$ + 5//2>
Norm: 2
Minimum: 2
two normal wrt: 2

julia> order(c[1])
9

julia> mc(c[1])^Int(order(c[1]))
<512, 43959719112139289493//2*_$^2 - 21814811847856022656*_$ + 47593247393471446019//2>
Norm: 512
Minimum: 512
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 — Method

is_principal(A::AbsSimpleNumFieldOrderIdeal) -> Bool
is_principal(A::AbsSimpleNumFieldOrderFractionalIdeal) -> Bool

Tests if $A$ is principal.

is_principal_with_data — Method

is_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 — Method

is_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 — Method

power_class(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, e::ZZRingElem) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

Computes a (small) ideal in the same class as $A^e$.

power_product_class — Method

power_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 — Method

power_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 — Method

class_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 — Method

factor_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 — Method

factor_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 — Function

prime_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, 3//2*_$^2 + 2*_$ + 5//2>
Norm: 2
Minimum: 2
two normal wrt: 2

julia> is_principal(I)
false

julia> I = I^Int(order(c[1]))
<512, 43959719112139289493//2*_$^2 - 21814811847856022656*_$ + 47593247393471446019//2>
Norm: 512
Minimum: 512
two normal wrt: 2

julia> is_principal(I)
true

julia> is_principal_fac_elem(I)
(true, (_$ + 29)^2*5^-3*(1//2*_$^2 - 6*_$ + 5//2)^2*(1//2*_$^2 - 6*_$ + 7//2)^2*31^-2*(19//2*_$^2 - 117*_$ + 117//2)^2*(_$ - 4)^-2*(_$^2 + _$ + 2)^3*(_$ + 5)^-3*(_$^2 + 1)^-3*3^1*(10*_$^2 - 91*_$ - 156)^-2*(_$ + 1)^1*(3//2*_$^2 + 1//2)^2*(_$ + 6)^2*1^-1)

The computation of $S$-units is also tied to the class group:

torsion_units — Method

torsion_units(O::AbsSimpleNumFieldOrder) -> Vector{AbsSimpleNumFieldOrderElem}

Given an order $O$, compute the torsion units of $O$.

torsion_unit_group — Method

torsion_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 — Method

torsion_units_generator(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem

Given an order $O$, compute a generator of the torsion units of $O$.

torsion_units_gen_order — Method

torsion_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 — Method

unit_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 — Method

unit_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 — Method

sunit_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 — Method

sunit_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 — Method

sunit_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
)

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])
(-1//2*_$^2 + 6*_$ - 3//2)^-1*(_$^2 + 1)^1*(_$ + 5)^1*3^-1*2^-2

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, 1//2*_$^2 + 3//2>        => 1

julia> s, ms = Hecke.sunit_group(collect(keys(lp)))
(Z/2 x Z^(5), SUnits  map of k for AbsSimpleNumFieldOrderIdeal[<3, _$ + 5>, <3, _$^2 + 1>, <2, 7//2*_$^2 + 2*_$ + 7//2>, <2, 1//2*_$^2 + 3//2>]
)

julia> ms(s[4])
1//2*_$^2 - 6*_$ - 5//2

julia> norm(ans)
-144

julia> factor(numerator(ans))
-1 * 2^4 * 3^2

Miscellaneous

order — Method

order(::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(K::AbsSimpleNumField, A::ZZMatrix, check::Bool = true) -> AbsSimpleNumFieldOrder

Returns the order which has basis matrix $A$ with respect to the power basis of $K$. If check is set, it is checked whether $A$ defines an order.

order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

order(A::AbstractAssociativeAlgebra{<: NumFieldElem}, M::PMat{<: NumFieldElem, T})
  -> AlgAssRelOrd

Returns the order of $A$ with basis pseudo-matrix $M$.

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(W::WeylGroup) -> ZZRingELem
order(::Type{T}, W::WeylGroup) where {T} -> T

Return the order of W.

If W is infinite, an InfiniteOrderError exception will be thrown. Use is_finite(::WeylGroup) to check this prior to calling this function if in doubt.

order(x::WeylGroupElem) -> ZZRingELem
order(::Type{T}, x::WeylGroupElem) where {T} -> T

Return the order of x, i.e. the smallest natural number n such that is_one(x^n).

If x is of infinite order, an InfiniteOrderError exception will be thrown. Use is_finite_order(::WeylGroupElem) to check this prior to calling this function if in doubt.

order — Method

order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

The order that was used to define the ideal $a$.

order — Method

order(::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(K::AbsSimpleNumField, A::ZZMatrix, check::Bool = true) -> AbsSimpleNumFieldOrder

Returns the order which has basis matrix $A$ with respect to the power basis of $K$. If check is set, it is checked whether $A$ defines an order.

order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

order(A::AbstractAssociativeAlgebra{<: NumFieldElem}, M::PMat{<: NumFieldElem, T})
  -> AlgAssRelOrd

Returns the order of $A$ with basis pseudo-matrix $M$.

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(W::WeylGroup) -> ZZRingELem
order(::Type{T}, W::WeylGroup) where {T} -> T

Return the order of W.

If W is infinite, an InfiniteOrderError exception will be thrown. Use is_finite(::WeylGroup) to check this prior to calling this function if in doubt.

order(x::WeylGroupElem) -> ZZRingELem
order(::Type{T}, x::WeylGroupElem) where {T} -> T

Return the order of x, i.e. the smallest natural number n such that is_one(x^n).

If x is of infinite order, an InfiniteOrderError exception will be thrown. Use is_finite_order(::WeylGroupElem) to check this prior to calling this function if in doubt.

order — Method

order(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrder

Returns the order of $a$.

nf — Method

nf(x::NumFieldOrderIdeal) -> AbsSimpleNumField

Returns the number field, of which $x$ is an integral ideal.

basis — Method

basis(A::AbsNumFieldOrderIdeal) -> Vector{AbsSimpleNumFieldOrderElem}

Returns the basis of $A$.

basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}

Returns the $\mathbf Z$-basis of $I$.

Return the default basis.

If (xᵢ)ᵢ is the basis of the distinguished cyclic subfield, and π the generating element, this basis is colex ordered (xᵢ⋅πʲ)ᵢⱼ.

lll_basis — Method

lll_basis(I::NumFieldOrderIdeal) -> Vector{NumFieldElem}

A basis for $I$ that is reduced using the LLL algorithm for the Minkowski metric.

basis_matrix — Method

basis_matrix(A::AbsNumFieldOrderIdeal) -> ZZMatrix

Returns the basis matrix of $A$.

basis_mat_inv — Method

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

Return the inverse of the basis matrix of $A$.

has_princ_gen_special — Method

has_princ_gen_special(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows if it is generated by a rational integer.

principal_generator — Method

principal_generator(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem

For a principal ideal $A$, find a generator.

principal_generator_fac_elem — Method

principal_generator_fac_elem(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> FacElem{AbsSimpleNumFieldElem, number_field}

For a principal ideal $A$, find a generator in factored form.

minimum — Method

minimum(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 — Method

minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

Returns the smallest non-negative element in $A \cap \mathbf Z$.

has_minimum — Method

has_minimum(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its minimum.

norm — Method

norm(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 — Method

has_norm(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its norm.

idempotents — Method

idempotents(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 — Method

is_prime(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool

Returns whether $A$ is a prime ideal.

is_prime_known — Method

is_prime_known(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool

Returns whether $A$ knows if it is prime.

is_ramified — Method

is_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 — Method

ramification_index(P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int

The ramification index of the prime-ideal $P$.

degree — Method

degree(P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int

The inertia degree of the prime-ideal $P$.

valuation — Method

valuation(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 — Method

valuation(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 — Method

valuation(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 — Method

valuation(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 — Method

valuation(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 — Method

valuation(A::AbsNumFieldOrderFractionalIdeal, p::AbsNumFieldOrderIdeal)

The valuation of $A$ at $p$.

idempotents — Method

idempotents(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 — Method

quo(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 projection $M: O\to O/I$. The pointwise inverse of $M$ implements a preimage/ lift function. In general this will not be a section as it will not be linear.

residue_ring — Method

residue_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 — Method

residue_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 — Method

mod(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 — Method

crt(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 — Method

euler_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 — Method

multiplicative_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 — Method

multiplicative_group_generators(Q::AbsSimpleNumFieldOrderQuoRing) -> Vector{AbsSimpleNumFieldOrderQuoRingElem}

Return a set of generators for $Q^\times$.