Fractional ideals

A fractional ideal in the number field $K$ is a $Z_K$-module $A$ such that there exists an integer $d>0$ which $dA$ is an (integral) ideal in $Z_K$. Due to the Dedekind property of $Z_K$, the ideals for a multiplicative group.

Fractional ideals are represented as an integral ideal and an additional denominator. They are of type AbsSimpleNumFieldOrderFractionalIdeal.

Creation

fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal

Creates the fractional ideal of $\mathcal{O}$ with basis matrix $M/b$. If M_in_hnf is set, then it is assumed that $A$ is already in lower left HNF.

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fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal

Creates the fractional ideal of $\mathcal{O}$ with basis matrix $M/b$. If M_in_hnf is set, then it is assumed that $A$ is already in lower left HNF.

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fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, M::QQMatrix) -> AbsNumFieldOrderFractionalIdeal

Creates the fractional ideal of $\mathcal{O}$ generated by the elements corresponding to the rows of $M$.

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fractional_ideal Method

fractional_ideal(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal

The fractional ideal of $O$ generated by a $Z$-basis of $I$.

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fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrderFractionalIdeal

Turns the ideal $I$ into a fractional ideal of $\mathcal{O}$.

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fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal, b::ZZRingElem) -> AbsNumFieldOrderFractionalIdeal

Creates the fractional ideal $I/b$ of $\mathcal{O}$.

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fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, a::RingElement) -> AbsNumFieldOrderFractionalIdeal

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

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fractional_ideal Method

fractional_ideal(O::AbsNumFieldOrder, a::RingElement) -> AbsNumFieldOrderFractionalIdeal

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

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inv Method

inv(A::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal

Computes the inverse of $A$, that is, the fractional ideal $B$ such that $AB={\mathcal{O}}_K$.

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Arithmetic

All the normal operations are provided as well.

inv Method

inv(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderFractionalIdeal

Returns the fractional ideal $B$ such that $AB=\mathcal{O}$.

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integral_split Method

integral_split(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderIdeal, AbsNumFieldOrderIdeal

Computes the unique coprime integral ideals $N$ and $D$ s.th. $A=N{D}^{-1}$

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numerator Method

numerator(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrderIdeal

Returns the ideal $d\ast a$ where $d$ is the denominator of $a$.

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denominator Method

denominator(a::RelNumFieldOrderFractionalIdeal) -> ZZRingElem

Returns the smallest positive integer $d$ such that $da$ is contained in the order of $a$.

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Miscellaneous

order Method

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

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

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order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

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

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

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

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basis_matrix Method

basis_matrix(I::AbsNumFieldOrderFractionalIdeal) -> FakeFmpqMat

Returns the basis matrix of $I$ with respect to the basis of the order.

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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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basis Method

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

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

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norm Method

norm(I::AbsNumFieldOrderFractionalIdeal) -> QQFieldElem

Returns the norm of $I$.

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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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