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.

source

# ideal — Method.

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

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

source

# ideal — Method.

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

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

source

# ideal — Method.

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

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

source

# 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, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal

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

source

# * — Method.

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

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

source

# 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).

source

# factor — Method.

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

Factors the principal ideal generated by $a$.

source

# 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.

source

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 $x\cap y$.

source

# 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.

source

# in — Method.

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

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

source

# 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.

source

# is_power — Method.

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

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

source

# 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.

source

# isone — Method.

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

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

source

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.

source

# 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.

source

# 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 $O$.

source

# ring_class_group — Method.

ring_class_group(O::AbsNumFieldOrder)

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

source

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}:
 [1]
 [4]
 [4]
 [5]
 [3]
 [2]
 [7]
 [1]
 [0]
 [2]

julia> mc(c[1])
<2, 5//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]))
<512, 1680483329278188243395//2*_$^2 - 833932253937892732350*_$ + 1819385119975802158297//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.

source

# 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.

source

# 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.

source

# power_class — Method.

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

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

source

# 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 $\prod A_i^{e_i}$.

source

# 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.

source

# 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 $c$.

source

# 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 $B$.

source

# 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 $B$.

source

# 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, 5//2*_$^2 + 7//2>
Norm: 2
Minimum: 2
two normal wrt: 2

julia> is_principal(I)
false

julia> I = I^Int(order(c[1]))
<512, 1680483329278188243395//2*_$^2 - 833932253937892732350*_$ + 1819385119975802158297//2>
Norm: 512
Minimum: 512
two normal wrt: 2

julia> is_principal(I)
true

julia> is_principal_fac_elem(I)
(true, 5^-1*(_$^2 + _$ + 2)^1*(_$ + 5)^-1*(_$^2 + 1)^-1*3^1*1^-1*(_$ - 3)^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 $O$.

source

# 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 $G\to {\mathcal{O}}^{\times }$.

source

# torsion_units_generator — Method.

torsion_units_generator(O::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrderElem

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

source

# 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.

source

# 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.

source

# 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.

source

# 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 $I$.

source

# 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.

source

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

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

julia> lp = factor(6*zk)
Dict{AbsSimpleNumFieldOrderIdeal, Int64} with 4 entries:
  <3, _$ + 5>                  => 1
  <3, _$^2 + 1>                => 1
  <2, 3//2*_$^2 + 7//2>        => 2
  <2, 3//2*_$^2 + 3*_$ + 7//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, _$^2 + 1>
Norm: 9
Minimum: 3
basis_matrix
[3 0 0; 0 3 0; 0 0 1]
two normal wrt: 3, <2, 3//2*_$^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*_$^2 + 3*_$ + 7//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 $I$.

source

# order — Method.

order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

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

source

# order — Method.

order(I::NumFieldOrderIdeal) -> AbsSimpleNumFieldOrder

Returns the order of $I$.

source

# order — Method.

order(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrder

Returns the order of $a$.

source

# nf — Method.

nf(x::NumFieldOrderIdeal) -> AbsSimpleNumField

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

source

# basis — Method.

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

Returns the basis of $A$.

source

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

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

source

# lll_basis — Method.

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

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

source

# basis_matrix — Method.

basis_matrix(A::AbsNumFieldOrderIdeal) -> ZZMatrix

Returns the basis matrix of $A$.

source

# basis_mat_inv — Method.

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

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

source

# has_princ_gen_special — Method.

has_princ_gen_special(A::AbsNumFieldOrderIdeal) -> Bool

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

source

# principal_generator — Method.

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

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

source

# 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.

source

# 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 $A$.

source

# minimum — Method.

minimum(A::AbsNumFieldOrderIdeal) -> ZZRingElem

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

source

# has_minimum — Method.

has_minimum(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its minimum.

source

# 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 $A$.

source

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

Returns the norm of $a$.

source

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

Returns the norm of $a$.

source

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

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

source

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 $O$.

source

# has_norm — Method.

has_norm(A::AbsNumFieldOrderIdeal) -> Bool

Returns whether $A$ knows its norm.

source

# 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.

source

# is_prime — Method.

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

Returns whether $A$ is a prime ideal.

source

# is_prime_known — Method.

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

Returns whether $A$ knows if it is prime.

source

# 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.

source

# ramification_index — Method.

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

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

source

# degree — Method.

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

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

source

# 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 ${\mathfrak{p}}^i$.

source

# 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 ${\mathfrak{p}}^i$.

source

# 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 ${\mathfrak{p}}^i$.

source

# 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 ${\mathfrak{p}}^i$.

source

# 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 ${\mathfrak{p}}^i$.

source

# valuation — Method.

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

The valuation of $A$ at $p$.

source

# 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.

source

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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