abelian_closure.md

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Abelian closure of the rationals

The abelian closure $\mathbf{Q}^\text{ab}$ is the maximal abelian extension of $\mathbf{Q}$ inside a fixed algebraic closure and can explicitly described as

$$\mathbf{Q}^{\mathrm{ab}} = \mathbf{Q}(\zeta_n \mid n \in \mathbf{N}),$$

the union of all cyclotomic extensions. Here for $n \in \mathbf{N}$ we denote by $\zeta_n$ the primitive $n$-th root of unity that corresponds to $\exp(2\pi i/n)$.

Creation of the abelian closure and elements

abelian_closure(::QQField)

Given the abelian closure, the generator can be recovered as follows:

gen(::QQAbField{AbsSimpleNumField})
atlas_irrationality
atlas_description

Natural embedding

OSCAR assumes a natural embedding of the field K produced by K, z = abelian_closure(QQ) into F = algebraic_closure(QQ), which is given by mapping the n-th root of unity returned by z(n) to root_of_unity(F, n). Both roots of unity correspond to the complex number $\exp(2 \pi i / n)$.

We can convert elements of K to elements of F as follows.

julia> K, z = abelian_closure(QQ)
(Abelian closure of rational field, Generator of abelian closure of rational field)

julia> F = algebraic_closure(QQ)
Algebraic closure of rational field

julia> x = z(5)
zeta(5)

julia> y = F(x)
{a4: 0.309017 + 0.951057*im}

julia> y^5
{a1: 1.00000}

Real elements of K can be compared with < and >.

julia> a = x + x^4
-zeta(5)^3 - zeta(5)^2 - 1

julia> a > 0
true

Printing

The n-th primitive root of the abelian closure of $\mathbf{Q}$ will by default be printed as zeta(n). The printing can be manipulated using the following functions:

gen(::QQAbField, ::String)
set_variable!(::QQAbField, ::String)
get_variable(::QQAbField)

Examples

julia> K, z = abelian_closure(QQ);

julia> z(4)
zeta(4)

julia> ζ = gen(K, "ζ")
Generator of abelian closure of QQ

julia> ζ(5) + ζ(3)
ζ(15)^5 + ζ(15)^3

julia> z(4)
"ζ"

julia> set_variable!(K, "zeta");

julia> z(4)
zeta(4)

Reduction to characteristic p

reduce(val::QQAbFieldElem, F::FinField)