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::AbsSimpleNumFieldElem) -> AbsNumFieldOrderFractionalIdeal

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

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# fractional_ideal — Method.

fractional_ideal(O::AbsNumFieldOrder, a::AbsNumFieldOrderElem) -> 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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Miscaellenous

# order — Method.

order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder

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

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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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