# 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

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

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

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

Factors the principal ideal generated by $a$.

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

Returns $x \cap y$.

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colonMethod
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.

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inMethod
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_powerMethod
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_powerMethod
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_invertibleMethod
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.

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isoneMethod
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_groupMethod
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_groupMethod
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_groupMethod
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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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*_$^2 + 2*_$ + 1//2>
Norm: 2
Minimum: 2
two normal wrt: 2julia> order(c[1])9julia> mc(c[1])^Int(order(c[1]))<32, 10991426517025591//2*_$^2 - 5454445711922803*_$ + 11899932301596121//2>
Norm: 512
Minimum: 32
two normal wrt: 2julia> mc \ ansAbelian 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_principalMethod
is_principal(A::AbsSimpleNumFieldOrderIdeal) -> Bool
is_principal(A::AbsSimpleNumFieldOrderFractionalIdeal) -> Bool

Tests if $A$ is principal.

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is_principal_with_dataMethod
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.

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is_principal_fac_elemMethod
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.

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power_classMethod
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_classMethod
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_reduceMethod
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_relationMethod
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_grhMethod
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_bachMethod
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_toFunction
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.

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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 $p$.

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julia> I = mc(c[1])<2, 1//2*_$^2 + 2*_$ + 1//2>
Norm: 2
Minimum: 2
two normal wrt: 2julia> is_principal(I)falsejulia> I = I^Int(order(c[1]))<32, 10991426517025591//2*_$^2 - 5454445711922803*_$ + 11899932301596121//2>
Norm: 512
Minimum: 32
two normal wrt: 2julia> is_principal(I)truejulia> is_principal_fac_elem(I)(true, 5^1*(1//2*_$^2 - 6*_$ + 5//2)^4*(_$- 3)^-1*(_$^2 + _$+ 2)^-1*(_$ + 5)^2*(_$^2 + 1)^2*3^-2*11^-1*(_$ + 1)^-1*(_$+ 2)^-2*1^-1) The computation of$S$-units is also tied to the class group: torsion_unitsMethod torsion_units(O::AbsSimpleNumFieldOrder) -> Vector{AbsSimpleNumFieldOrderElem} Given an order$O$, compute the torsion units of$O$. source torsion_unit_groupMethod torsion_unit_group(O::AbsSimpleNumFieldOrder) -> GrpAb, Map Given an order$\mathcal O$, returns the torsion units as an abelian group$G$together with a map$G \to \mathcal O^\times$. source torsion_units_generatorMethod torsion_units_generator(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem Given an order$O$, compute a generator of the torsion units of$O$. source torsion_units_gen_orderMethod torsion_units_gen_order(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem Given an order$O$, compute a generator of the torsion units of$O$as well as its order. source unit_groupMethod unit_group(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map Returns a group$U$and an isomorphism map$f \colon U \to \mathcal O^\times$. A set of fundamental units of$\mathcal O$can be obtained via [ f(U[1+i]) for i in 1:unit_group_rank(O) ]. f(U[1]) will give a generator for the torsion subgroup. source unit_group_fac_elemMethod unit_group_fac_elem(O::AbsSimpleNumFieldOrder) -> FinGenAbGroup, Map Returns a group$U$and an isomorphism map$f \colon U \to \mathcal O^\times$. A set of fundamental units of$\mathcal O$can be obtained via [ f(U[1+i]) for i in 1:unit_group_rank(O) ]. f(U[1]) will give a generator for the torsion subgroup. All elements will be returned in factored form. source sunit_groupMethod sunit_group(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map For an array$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$. source sunit_group_fac_elemMethod sunit_group_fac_elem(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map For an array$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$. The map will return elements in factored form. source sunit_mod_units_group_fac_elemMethod sunit_mod_units_group_fac_elem(I::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> GrpAb, Map For an array$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 + _$- 1julia> u, mu = unit_group_fac_elem(zk)(Z/2 x Z, UnitGroup map of Factored elements over Number field of degree 3 over QQ )julia> mu(u[2])(-6*_$^2 + 76*_$+ 32)^-1*(3//2*_$^2 + 5*_$- 423//2)^1*(_$^2 + 1)^1*(19//2*_$^2 - 135*_$ + 341//2)^-1*(1//2*_$^2 - 6*_$ + 7//2)^-1*(_$+ 2)^1*(_$ + 11)^1julia> evaluate(ans)-_$^2 + _$ - 1julia> lp = factor(6*zk)Dict{AbsSimpleNumFieldOrderIdeal, Int64} with 4 entries:
<3, _$+ 5> => 1 <3, _$^2 + 1>                => 1
<2, 7//2*_$^2 + 2*_$ + 3//2> => 2
<2, 5//2*_$^2 + 2*_$ + 3//2> => 1julia> 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*_$^2 + 2*_$ + 3//2>
Norm: 2
Minimum: 2
basis_matrix
[2 0 0; 1 1 0; 0 0 1]
two normal wrt: 2, <2, 5//2*_$^2 + 2*_$ + 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//2julia> norm(ans)-144julia> factor(numerator(ans))-1 * 2^4 * 3^2

## Miscaellenous

orderMethod
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.

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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.

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order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

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

Returns the order of W.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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orderMethod
order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

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

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orderMethod
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.

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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.

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order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

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

Returns the order of W.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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orderMethod
order(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrder

Returns the order of $a$.

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nfMethod
nf(x::NumFieldOrderIdeal) -> AbsSimpleNumField

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

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

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

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principal_generator_fac_elemMethod
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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minimumMethod
minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

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

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  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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minimumMethod
  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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minimumMethod
minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

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

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normMethod
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_normMethod
has_norm(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its norm.

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

Returns whether $A$ is a prime ideal.

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

Returns whether $A$ knows if it is prime.

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

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

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

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

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

The valuation of $A$ at $p$.

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idempotentsMethod
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

quoMethod
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_ringMethod
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_fieldMethod
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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modMethod
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 $1 \leq i \leq d$.

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crtMethod
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.

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euler_phiMethod
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_groupMethod
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 $f \colon A \to Q^\times$.

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