Fractional ideals

Fractional ideals

A fractional ideal in the number field $K$ is a $Z_K$-module $A$ such that there exists an integer $d>0$ wich $dA$ 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 NfOrdFracIdl.

Creation

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, A::fmpz_mat, b::fmpz, A_in_hnf::Bool = false) -> NfOrdFracIdl

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

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, A::fmpz_mat, b::fmpz, A_in_hnf::Bool = false) -> NfOrdFracIdl

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

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, A::FakeFmpqMat, A_in_hnf::Bool = false) -> NfOrdFracIdl

Creates the fractional ideal of $\mathcal O$ with basis matrix $A$. If Ainhnf is set, then it is assumed that the numerator of $A$ is already in lower left HNF.

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, I::NfOrdIdl) -> NfOrdFracIdl

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

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, I::NfOrdIdl, b::fmpz) -> NfOrdFracIdl

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

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, a::nf_elem) -> NfOrdFracIdl

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

Hecke.frac_ideal — Method.
frac_ideal(O::NfOrd, a::NfOrdElem) -> NfOrdFracIdl

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

Base.inv — Method.
inv(A::NfAbsOrdIdl) -> NfOrdFracIdl

Computes the inverse of A, that is, the fractional ideal $B$ such that $AB = \mathcal O_K$.

  inv(a::NfRelOrdIdl) -> NfRelOrdFracIdl
  inv(a::NfRelOrdFracIdl) -> NfRelOrdFracIdl

Computes the inverse of $a$, that is, the fractional ideal $b$ such that $ab = O$, where $O$ is the ambient order of $a$. $O$ must be maximal.

Arithmetic

Base.:== — Method.
==(x::NfOrdFracIdl, y::NfOrdFracIdl) -> Bool

Returns whether $x$ and $y$ are equal.

Base.inv — Method.
inv(A::NfOrdFracIdl) -> NfOrdFracIdl

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

Hecke.integral_split — Method.
integral_split(A::NfOrdFracIdl) -> NfOrdIdl, NfOrdIdl

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

Miscaellenous

AbstractAlgebra.Generic.order — Method.
order(a::NfOrdFracIdl) -> NfOrd

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

Hecke.basis_mat — Method.
basis_mat(I::NfOrdFracIdl) -> FakeFmpqMat

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

Hecke.basis_mat_inv — Method.
basis_mat_inv(I::NfOrdFracIdl) -> FakeFmpqMat

Returns the inverse of the basis matrix of $I$.

Hecke.basis — Method.
basis(I::NfOrdFracIdl) -> Array{nf_elem, 1}

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

LinearAlgebra.norm — Method.
norm(I::NfOrdFracIdl) -> fmpq

Returns the norm of $I$