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 $a$.

source

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.

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ideal Method

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

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

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ideal Method

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

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

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ideal Method

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

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

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ideal Method

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

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

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ideal Method

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

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

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* Method

*(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
*(x::AbsNumFieldOrderElem, O::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal

Returns the principal ideal $(x)$ of $\mathcal{O}$.

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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).

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factor Method

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

Factors the principal ideal generated by $a$.

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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.

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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 $x\cap y$.

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colon Method

colon(a::AbsNumFieldOrderIdeal, b::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal

The ideal $(a:b)=\{x\in K|xb\subseteq a\}=\mathrm{hom}(b,a)$ where $K$ is the number field.

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in Method

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

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

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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.

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is_power Method

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

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

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is_invertible Method

is_invertible(A::AbsNumFieldOrderIdeal) -> Bool, AbsSimpleNumFieldOrderFractionalIdeal

Returns true and an inverse of $A$ or false and an ideal $B$ such that $A\ast B⊊order(A)$, if $A$ is not invertible.

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isone Method

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

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

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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 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.

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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.

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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 $O$.

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ring_class_group Method

ring_class_group(O::AbsNumFieldOrder)

The ring class group (Picard group) of $O$.

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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, 1//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]))
<32, 13217457633644257//2*_$^2 - 6559103588818597*_$ + 14309957929619471//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 Method

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

Tests if $A$ is principal.

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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 $A=⟨\alpha ⟩$ or $(\mathtt{false},1)$ otherwise.

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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 $A=⟨\alpha ⟩$ or $(\mathtt{false},1)$ otherwise. The generator will be in factored form.

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power_class Method

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

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

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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 $\prod A_i^{e_i}$.

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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.

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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 $c$.

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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 $B$.

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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 $B$.

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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 $\mathrm{deg}(\mathfrak{p})>k$ will be discarded. If 'index_divisors' is set to false, only primes not dividing the index of the order will be computed.

source

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 $\mathrm{deg}(\mathfrak{p})>k$ will be discarded.

The function $F$ must be a function on prime numbers not dividing bad such that $F(p)=\mathrm{deg}(\mathfrak{p})$ for all prime ideals $\mathfrak{p}$ lying above $p$.

source

julia> I = mc(c[1])
<2, 1//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]))
<32, 13217457633644257//2*_$^2 - 6559103588818597*_$ + 14309957929619471//2>
Norm: 512
Minimum: 32
two normal wrt: 2

julia> is_principal(I)
true

julia> is_principal_fac_elem(I)
(true, (1//2*_$^2 - 3*_$ + 29//2)^-4*5^-3*(_$ - 3)^-1*(_$^2 + _$ + 2)^3*(_$ + 5)^-2*(_$^2 + 1)^-2*3^2*(4*_$^2 - 46*_$ - 6)^4*11^-1*(_$ + 1)^3*(_$ + 2)^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 $O$.

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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 $G\to {\mathcal{O}}^{\times }$.

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torsion_units_generator Method

torsion_units_generator(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem

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

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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.

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unit_group Method

unit_group(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map

Returns a group $U$ and an isomorphism map $f:U\to {\mathcal{O}}^{\times }$. 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.

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unit_group_fac_elem Method

unit_group_fac_elem(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map

Returns a group $U$ and an isomorphism map $f:U\to {\mathcal{O}}^{\times }$. 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.

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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 $I$.

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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.

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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.

source

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])
(_$^2 + 1)^1*(6*_$^2 - 64*_$ + 5)^1*(1//2*_$^2 - 6*_$ + 7//2)^-1*(15//2*_$^2 - 81*_$ - 155//2)^-1

julia> evaluate(ans)
_$^2 - _$ + 1

julia> lp = factor(6*zk)
Dict{AbsSimpleNumFieldOrderIdeal, Int64} with 4 entries:
  <3, _$ + 5>                  => 1
  <3, _$^2 + 1>                => 1
  <2, 1//2*_$^2 + 2*_$ + 1//2> => 2
  <2, 3//2*_$^2 + 1//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, _$^2 + 1>
Norm: 9
Minimum: 3
basis_matrix
[3 0 0; 0 3 0; 0 0 1]
two normal wrt: 3, <2, 1//2*_$^2 + 2*_$ + 1//2>
Norm: 2
Minimum: 2
basis_matrix
[2 0 0; 1 1 0; 0 0 1]
two normal wrt: 2, <2, 3//2*_$^2 + 1//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 Method

order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

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order Method

order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

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

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order Method

order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

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order Method

order(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrder

Returns the order of $a$.

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nf Method

nf(x::NumFieldOrderIdeal) -> AbsSimpleNumField

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

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basis Method

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

Returns the basis of $A$.

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

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

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lll_basis Method

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

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

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basis_matrix Method

basis_matrix(A::AbsNumFieldOrderIdeal) -> ZZMatrix

Returns the basis matrix of $A$.

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basis_mat_inv Method

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

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

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has_princ_gen_special Method

has_princ_gen_special(A::AbsNumFieldOrderIdeal) -> Bool

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

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principal_generator Method

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

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

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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.

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minimum Method

minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

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

source

  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 $A$.

source

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 $A$.

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minimum Method

minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

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

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has_minimum Method

has_minimum(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its minimum.

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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 $A$.

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norm(a::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

Returns the norm of $a$.

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norm(a::RelNumFieldOrderFractionalIdeal{T, S}) -> S

Returns the norm of $a$.

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norm(a::AlgAssAbsOrdIdl, O::AlgAssAbsOrd; copy::Bool = true) -> QQFieldElem

Returns the norm of $a$ considered as an (possibly fractional) ideal of $O$.

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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 $O$.

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has_norm Method

has_norm(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its norm.

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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.

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is_prime Method

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

Returns whether $A$ is a prime ideal.

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is_prime_known Method

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

Returns whether $A$ knows if it is prime.

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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.

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ramification_index Method

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

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

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degree Method

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

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

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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 ${\mathfrak{p}}^i$.

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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 ${\mathfrak{p}}^i$.

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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 ${\mathfrak{p}}^i$.

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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 ${\mathfrak{p}}^i$.

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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 ${\mathfrak{p}}^i$.

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valuation Method

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

The valuation of $A$ at $p$.

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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.

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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 section $M:O/I\to O$. The pointwise inverse of $M$ is the canonical projection $O\to O/I$.

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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.

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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.

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mod Method

mod(x::AbsSimpleNumFieldOrderElem, I::AbsNumFieldOrderIdeal)

Returns the unique element $y$ of the ambient order of $x$ with $x\equiv ymodI$ and the following property: If $a_1,\dots ,a_d\in {\mathbf{Z}}_{\ge 1}$ are the diagonal entries of the unique HNF basis matrix of $I$ and $(b_1,\dots ,b_d)$ is the coefficient vector of $y$, then $0\le b_i<a_i$ for $1\le i\le d$.

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crt Method

crt(r1::AbsSimpleNumFieldOrderElem, i1::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, r2::AbsSimpleNumFieldOrderElem, i2::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem

Find $x$ such that $x\equiv r_{1}mod{i}_1$ and $x\equiv r_{2}mod{i}_2$ using idempotents.

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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.

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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 $f:A\to Q^{\times }$.

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multiplicative_group_generators Method

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

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

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