Element operations

Creation

We can return the generator $\alpha$ of a simple extension $K(\alpha )/K$ and the vector of generators ${\alpha }_1,...,{\alpha }_n$ of a nonsimple extension $K({\alpha }_1,...,{\alpha }_n)/K$ with the following.

gen Method

gen(L::SimpleNumField) -> NumFieldElem

Given a simple number field $L=K[x]/(f)$ over $K$, this functions returns the class of $x$, which is the canonical primitive element of $L$ over $K$.

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gens Method

gens(L::NonSimpleNumField) -> Vector{NumFieldElem}

Given a non-simple number field $L=K[x_1,\dots ,x_n]/(f_1,\dots ,f_n)$ over $K$, this functions returns the list ${\overline{x}}_1,\dots ,{\overline{x}}_n$.

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Elements can also be created by specifying the coordinates with respect to the basis of the number field:

(L::number_field)(c::Vector{NumFieldElem}) -> NumFieldElem

Given a number field $L/K$ of degree $d$ and a vector c of elements from $K$ of length $d$, the above method constructs the element a with coordinates(a) == c.

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^2 - 2, "a");

julia> basis(K)
2-element Vector{AbsSimpleNumFieldElem}:
 1
 a

julia> K([1, 2])
2*a + 1

julia> L, b = radical_extension(3, a, "b")
(Relative number field of degree 3 over K, b)

julia> basis(L)
3-element Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}:
 1
 b
 b^2

julia> L([a, 1, 1//2])
1//2*b^2 + b + a

Conversely, given an element $x$ of a number field $K$, we can extract the coordinates of $x$ under various bases.

coordinates Method

coordinates(x::NumFieldElem{T}) -> Vector{T}

Given an element $x$ in a number field $K$, this function returns the coordinates of $x$ with respect to the basis of $K$ (the output of the 'basis' function).

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absolute_coordinates Method

absolute_coordinates(x::NumFieldElem{T}) -> Vector{T}

Given an element $x$ in a number field $K$, this function returns the coordinates of $x$ with respect to the basis of $K$ over the rationals (the output of the absolute_basis function).

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coefficients Method

coefficients(a::SimpleNumFieldElem) -> Vector{FieldElem}

Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.

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coeff Method

coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem

Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.

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Functions on elements

The following collections of functions all take as input an element of a number field, or a vector of such elements, possibly with some additional data.

Basis Dependent

Functions that depend on a basis:

representation_matrix Method

representation_matrix(a::NumFieldElem) -> MatElem

Returns the representation matrix of $a$, that is, the matrix representing multiplication with $a$ with respect to the canonical basis of the parent of $a$.

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basis_matrix Method

basis_matrix(v::Vector{NumFieldElem}) -> Mat

Given a vector $v$ of $n$ elements of a number field $K$ of degree $d$, this function returns an $n\times d$ matrix with entries in the base field of $K$, where row $i$ contains the coefficients of $v[i]$ with respect of the canonical basis of $K$.

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Invariants

Common invariants of an element:

norm Method

norm(a::NumFieldElem) -> NumFieldElem

Returns the norm of an element $a$ of a number field extension $L/K$. This will be an element of $K$.

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absolute_norm Method

absolute_norm(a::NumFieldElem) -> QQFieldElem

Given a number field element $a$, returns the absolute norm of $a$.

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norm Method

norm(a::NumFieldElem, k::NumField) -> NumFieldElem

Returns the norm of an element $a$ of a number field $L$ with respect to a subfield $k$ of $L$. This will be an element of $k$.

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tr Method

tr(a::NumFieldElem) -> NumFieldElem

Returns the trace of an element $a$ of a number field extension $L/K$. This will be an element of $K$.

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absolute_tr Method

absolute_tr(a::NumFieldElem) -> QQFieldElem

Given a number field element $a$, returns the absolute trace of $a$.

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minpoly Method

minpoly(a::NumFieldElem) -> PolyRingElem

Given a number field element $a$ of a number field $K$, this function returns the minimal polynomial of $a$ over the base field of $K$.

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absolute_minpoly Method

absolute_minpoly(a::NumFieldElem) -> PolyRingElem

Given a number field element $a$ of a number field $K$, this function returns the minimal polynomial of $a$ over the rationals $\mathbf{Q}$.

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charpoly Method

charpoly(a::NumFieldElem) -> PolyRingElem

Given a number field element $a$ of a number field $K$, this function returns the characteristic polynomial of $a$ over the base field of $K$.

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absolute_charpoly Method

absolute_charpoly(a::NumFieldElem) -> PolyRingElem

Given a number field element $a$ of a number field $K$, this function returns the characteristic polynomial of $a$ over the rationals $\mathbf{Q}$.

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Predicates

is_integral Method

is_integral(a::NumFieldElem) -> Bool

Returns whether $a$ is integral, that is, whether the minimal polynomial of $a$ has integral coefficients.

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is_torsion_unit Method

is_torsion_unit(x::AbsSimpleNumFieldElem, checkisunit::Bool = false) -> Bool

Returns whether $x$ is a torsion unit, that is, whether there exists $n$ such that $x^n=1$.

If checkisunit is true, it is first checked whether $x$ is a unit of the maximal order of the number field $x$ is lying in.

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is_local_norm Method

is_local_norm(L::NumField, a::NumFieldElem, P)

Given a number field $L/K$, an element $a\in K$ and a prime ideal $P$ of $K$, returns whether $a$ is a local norm at $P$.

The number field $L/K$ must be a simple extension of degree 2.

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is_norm_divisible Method

is_norm_divisible(a::AbsSimpleNumFieldElem, n::ZZRingElem) -> Bool

Checks if the norm of $a$ is divisible by $n$, assuming that the norm of $a$ is an integer.

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is_norm Method

is_norm(K::AbsSimpleNumField, a) -> Bool, AbsSimpleNumFieldElem

For $a$ an integer or rational, try to find $T\in K$ s.th. $N(T)=a$ holds. If successful, return true and $T$, otherwise false and some element.

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Conjugates

Given an absolute simple number field $K$, the signature of $K$ is a pair of integers $(r,s)$ such that $K$ has $r$ real embeddings ${\sigma }_i:K\to \mathbf{R}$, $1\le i\le r$, and $2s$ complex embeddings ${\sigma }_{r+i}:K\to \mathbf{C}$, $1\le i\le 2s$. In Hecke the complex embeddings are always ordered such that ${\sigma }_i=\overline{{\sigma }_{i+s}}$ for $r+1\le i\le r+s$.

conjugates Method

conjugates(x::AbsSimpleNumFieldElem, C::AcbField) -> Vector{AcbFieldElem}

Compute the conjugates of $x$ as elements of type AcbFieldElem. Recall that we order the complex conjugates ${\sigma }_{r+1}(x),...,{\sigma }_{r+2s}(x)$ such that ${\sigma }_i(x)=\overline{{\sigma }_{i+s}(x)}$ for $r+1\le i\le r+s$.

Let p be the precision of C, then every entry $y$ of the vector returned satisfies radius(real(y)) < 2^-p and radius(imag(y)) < 2^-p respectively.

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conjugates Method

conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}

Compute the conjugates of $x$ as elements of type AcbFieldElem. Recall that we order the complex conjugates ${\sigma }_{r+1}(x),...,{\sigma }_{r+2s}(x)$ such that ${\sigma }_i(x)=\overline{{\sigma }_{i+s}(x)}$ for $r+1\le i\le r+s$.

Every entry $y$ of the vector returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol respectively.

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conjugates_log Method

conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}

Returns the elements $(\log (|{\sigma }_1(x)|),\dots ,\log (|{\sigma }_r(x)|),\dots ,2\log (|{\sigma }_{r+1}(x)|),\dots ,2\log (|{\sigma }_{r+s}(x)|))$ as elements of type ArbFieldElem with radius less then 2^-abs_tol.

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conjugates_real Method

conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}

Compute the real conjugates of $x$ as elements of type ArbFieldElem.

Every entry $y$ of the array returned satisfies radius(y) < 2^-abs_tol.

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conjugates_complex Method

conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}

Compute the complex conjugates of $x$ as elements of type AcbFieldElem. Recall that we order the complex conjugates ${\sigma }_{r+1}(x),...,{\sigma }_{r+2s}(x)$ such that ${\sigma }_i(x)=\overline{{\sigma }_{i+s}(x)}$ for $r+1\le i\le r+s$.

Every entry $y$ of the array returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol.

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conjugates_arb_log_normalise Method

conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
conjugates_arb_log_normalise(x::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, p::Int = 10)

The "normalised" logarithms, i.e. the array $c_i\log |x^{(i)}|-1/n\log |N(x)|$, so the (weighted) sum adds up to zero.

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The $\mathbf{Q}$-linear function

$$\begin{array}{c}K⟶{\mathbf{R}}^d\\ \alpha ⟼({\sigma }_1(\alpha ),\dots ,{\sigma }_r(\alpha ),\sqrt{2}\mathrm{Re}({\sigma }_{r+1}(\alpha )),\sqrt{2}\mathrm{Im}({\sigma }_{r+1}(\alpha )),\dots ,\sqrt{2}\mathrm{Re}({\sigma }_{r+s}(\alpha )),\sqrt{2}\mathrm{Im}({\sigma }_{r+s}(\alpha )))\end{array}$$

is called the Minkowski map (or Minkowski embedding). We provide a function that returns the image of an element under the Minkowski map.

minkowski_map Method

minkowski_map(a::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}

Returns the image of $a$ under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^(-abs_tol).

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Miscellaneous

quadratic_defect Method

quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}

Returns the valuation of the quadratic defect of the element $a$ at $p$, which can either be prime object or an infinite place of the parent of $a$.

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hilbert_symbol Method

hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int

Returns the local Hilbert symbol $(a,b{)}_p$.

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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 ${\mathfrak{p}}^i$.

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torsion_unit_order Method

torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)

Given a torsion unit $x$ together with a multiple $n$ of its order, compute the order of $x$, that is, the smallest $k\in {\mathbb{Z}}_{\ge 1}$ such that $x^k=1$.

It is not checked whether $x$ is a torsion unit.

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algebraic_split Method

algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem

Writes the input as a quotient of two "small" algebraic integers.

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