Orders

order(a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder
order(K::AbsSimpleNumField, a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder

Returns the order generated by $a$. If check is set, it is checked whether $a$ defines an order, in particular the integrality of the elements is checked by computing minimal polynomials. If isbasis is set, then elements are assumed to form a $\mathbf{Z}$-basis. If cached is set, then the constructed order is cached for future use.

order(K::AbsSimpleNumField, A::QQMatrix; 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(K::AbsSimpleNumField, A::QQMatrix; 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.

equation_order(K::number_field) -> NumFieldOrder
equation_order(K::number_field) -> NumFieldOrder

Returns the equation order of the number field $K$.

maximal_order(K::NumField{QQFieldElem}; discriminant::ZZRingElem, ramified_primes::Vector{ZZRingElem}) -> AbsNumFieldOrder

Returns the maximal order of $K$. Additional information can be supplied if they are already known, as the ramified primes or the discriminant of the maximal order.

Example

julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^3 + 2, "a");
julia> O = maximal_order(K);

maximal_order(O::AbsNumFieldOrder; index_divisors::Vector{ZZRingElem}, discriminant::ZZRingElem, ramified_primes::Vector{ZZRingElem}) -> AbsNumFieldOrder

Returns the maximal order of the number field that contains $O$. Additional information can be supplied if they are already known, as the ramified primes, the discriminant of the maximal order or a set of integers dividing the index of $O$ in the maximal order.

maximal_order(O::AlgAssAbsOrd)

Given an order $O$, this function returns a maximal order containing $O$.

maximal_order(A::AbstractAssociativeAlgebra{QQFieldElem}) -> AlgAssAbsOrd

Returns a maximal order of $A$.

maximal_order(O::AlgAssRelOrd) -> AlgAssRelOrd

Returns a maximal order of the algebra of $O$ containing itself.

lll(M::AbsNumFieldOrder) -> AbsNumFieldOrder

The same order, but with the basis now being LLL reduced wrt. the Minkowski metric.

any_order(K::number_field)

Return some order in $K$. In case the defining polynomial for $K$ is monic and integral, this just returns the equation order. In the other case $\mathbb Z[\alpha]\cap \mathbb Z[1/\alpha]$ is returned.

parent(O::AbsNumFieldOrder) -> AbsNumFieldOrderSet

Returns the parent of $\mathcal O$, that is, the set of orders of the ambient number field.

signature(O::NumFieldOrder) -> Tuple{Int, Int}

Returns the signature of the ambient number field of $\mathcal O$.

nf(O::NumFieldOrder) -> NumField

Returns the ambient number field of $\mathcal O$.

basis(O::AbsNumFieldOrder) -> Vector{AbsNumFieldOrderElem}

Returns the $\mathbf Z$-basis of $\mathcal O$.

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

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

lll_basis(M::NumFieldOrder) -> Vector{NumFieldElem}

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

basis(O::AbsSimpleNumFieldOrder, K::AbsSimpleNumField) -> Vector{AbsSimpleNumFieldElem}

Returns the $\mathbf Z$-basis elements of $\mathcal O$ as elements of the ambient number field.

  pseudo_basis(O::RelNumFieldOrder{T, S}) -> Vector{Tuple{NumFieldElem{T}, S}}

Returns the pseudo-basis of $\mathcal O$.

  basis_pmatrix(O::RelNumFieldOrder) -> PMat

Returns the basis pseudo-matrix of $\mathcal O$ with respect to the power basis of the ambient number field.

  basis_nf(O::RelNumFieldOrder) -> Vector{NumFieldElem}

Returns the elements of the pseudo-basis of $\mathcal O$ as elements of the ambient number field.

  inv_coeff_ideals(O::RelNumFieldOrder{T, S}) -> Vector{S}

Returns the inverses of the coefficient ideals of the pseudo basis of $O$.

basis_matrix(O::AbsNumFieldOrder) -> QQMatrix

Returns the basis matrix of $\mathcal O$ with respect to the basis of the ambient number field.

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

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

gen_index(O::AbsSimpleNumFieldOrder) -> QQFieldElem

Returns the generalized index of $\mathcal O$ with respect to the equation order of the ambient number field.

is_index_divisor(O::AbsSimpleNumFieldOrder, d::ZZRingElem) -> Bool
is_index_divisor(O::AbsSimpleNumFieldOrder, d::Int) -> Bool

Returns whether $d$ is a divisor of the index of $\mathcal O$. It is assumed that $\mathcal O$ contains the equation order of the ambient number field.

minkowski_matrix(O::AbsNumFieldOrder, abs_tol::Int = 64) -> ArbMatrix

Returns the Minkowski matrix of $\mathcal O$. Thus if $\mathcal O$ has degree $d$, then the result is a matrix in $\operatorname{Mat}_{d\times d}(\mathbf R)$. The entries of the matrix are real balls of type ArbFieldElem with radius less then 2^-abs_tol.

in(a::NumFieldElem, O::NumFieldOrder) -> Bool

Checks whether $a$ lies in $\mathcal O$.

norm_change_const(O::AbsSimpleNumFieldOrder) -> (Float64, Float64)

Returns $(c_1, c_2) \in \mathbf R_{>0}^2$ such that for all $x = \sum_{i=1}^d x_i \omega_i \in \mathcal O$ we have $T_2(x) \leq c_1 \cdot \sum_i^d x_i^2$ and $\sum_i^d x_i^2 \leq c_2 \cdot T_2(x)$, where $(\omega_i)_i$ is the $\mathbf Z$-basis of $\mathcal O$.

trace_matrix(O::AbsNumFieldOrder) -> ZZMatrix

Returns the trace matrix of $\mathcal O$, that is, the matrix $(\operatorname{tr}_{K/\mathbf Q}(b_i \cdot b_j))_{1 \leq i, j \leq d}$.

+(R::AbsSimpleNumFieldOrder, S::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrder

Given two orders $R$, $S$ of $K$, this function returns the smallest order containing both $R$ and $S$. It is assumed that $R$, $S$ contain the ambient equation order and have coprime index.

poverorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
poverorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder

This function tries to find an order that is locally larger than $\mathcal O$ at the prime $p$: If $p$ divides the index $[ \mathcal O_K : \mathcal O]$, this function will return an order $R$ such that $v_p([ \mathcal O_K : R]) < v_p([ \mathcal O_K : \mathcal O])$. Otherwise $\mathcal O$ is returned.

poverorders(O, p) -> Vector{Ord}

Returns all p-overorders of O, that is all overorders M, such that the index of O in M is a p-power.

pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder

This function finds a $p$-maximal order $R$ containing $\mathcal O$. That is, the index $[ \mathcal O_K : R]$ is not divisible by $p$.

pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal

Given a prime number $p$, this function returns the $p$-radical $\sqrt{p\mathcal O}$ of $\mathcal O$, which is just $\{ x \in \mathcal O \mid \exists k \in \mathbf Z_{\geq 0} \colon x^k \in p\mathcal O \}$. It is not checked that $p$ is prime.

  pradical(O::RelNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> RelNumFieldOrderIdeal

Given a prime ideal $P$, this function returns the $P$-radical $\sqrt{P\mathcal O}$ of $\mathcal O$, which is just $\{ x \in \mathcal O \mid \exists k \in \mathbf Z_{\geq 0} \colon x^k \in P\mathcal O \}$. It is not checked that $P$ is prime.

ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder

Computes the order $(I : I)$, which is the set of all $x \in K$ with $xI \subseteq I$.

discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem

Returns the discriminant of $\mathcal O$.

reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem

Returns the reduced discriminant, that is, the largest elementary divisor of the trace matrix of $\mathcal O$.

degree(O::NumFieldOrder) -> Int

Returns the degree of $\mathcal O$.

index(O::AbsSimpleNumFieldOrder) -> ZZRingElem

Assuming that the order $\mathcal O$ contains the equation order $\mathbf Z[\alpha]$ of the ambient number field, this function returns the index $[ \mathcal O : \mathbf Z]$.

different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal

The different ideal of $R$, that is, the ideal generated by all differents of elements in $R$. For Gorenstein orders, this is also the inverse ideal of the co-different.

codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

The codifferent ideal of $R$, i.e. the trace-dual of $R$.

is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool

Return whether the order \mathcal{O} is Gorenstein.

is_bass(O::AbsSimpleNumFieldOrder) -> Bool

Return whether the order \mathcal{O} is Bass.

is_equation_order(O::NumFieldOrder) -> Bool

Returns whether $\mathcal O$ is the equation order of the ambient number field $K$.

zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem

Computes the residue of the zeta function of $\mathcal O$ at $1$. The output will be an element of type ArbFieldElem with radius less then error.

ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}

Returns the list of prime numbers that divide $\operatorname{disc}(\mathcal O)$.

is_independent{T}(x::Vector{T})

Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.

is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool

Checks if $R$ is contained in $S$.

is_maximal(R::AbsNumFieldOrder) -> Bool

Tests if the order $R$ is maximal. This might trigger the computation of the maximal order.