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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 $\zeta_n$ a primitive $n$-th root of unity.

Creation of the abelian closure and elements

abelian_closure — Method
abelian_closure(QQ::QQField; sparse::Bool = false)

Return a pair (K, z) consisting of the abelian closure K of the rationals and a generator z that can be used to construct primitive roots of unity in K.

An optional keyword argument sparse can be set to true to switch to a sparse representation. Depending on the application this can be much faster or slower.

Examples

julia> K, z = abelian_closure(QQ);

julia> z(36)
zeta(36)

julia> K, z = abelian_closure(QQ, sparse = true);

julia> z(36)
-zeta(36, 9)*zeta(36, 4)^4 - zeta(36, 9)*zeta(36, 4)
source

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

gen — Method
gen(K::QQAbField)

Return the generator of the abelian closure K that can be used to construct primitive roots of unity.

source
atlas_irrationality — Function
atlas_irrationality([F::AbsSimpleNumField, ]description::String)

Return the value encoded by description. If F is given and is a cyclotomic field that contains the value then the result is in F, if F is not given then the result has type QQAbFieldElem.

description is assumed to have the format defined in [CCNPW85], Chapter 6, Section 10.

Examples

julia> Oscar.with_unicode() do
         show(atlas_irrationality("r5"))
       end;
-2*ζ(5)^3 - 2*ζ(5)^2 - 1

julia> atlas_irrationality(cyclotomic_field(5)[1], "r5")
-2*z_5^3 - 2*z_5^2 - 1

julia> Oscar.with_unicode() do
         show(atlas_irrationality("i"))
       end;
ζ(4)

julia> Oscar.with_unicode() do
         show(atlas_irrationality("b7*3"))
       end;
-ζ(7)^4 - ζ(7)^2 - ζ(7) - 1

julia> Oscar.with_unicode() do
         show(atlas_irrationality("3y'''24*13-2&5"))
       end;
-5*ζ(24)^7 - 2*ζ(24)^5 + 2*ζ(24)^3 - 3*ζ(24)
source
atlas_description — Function
atlas_description(val::QQAbFieldElem)

Return a string in the format defined in [CCNPW85], Chapter 6, Section 10, describing val. Applying atlas_irrationality to the result yields val.

Examples

julia> K, z = abelian_closure(QQ);

julia> val = z(5) + z(5)^4;

julia> str = Oscar.atlas_description(val)
"b5"

julia> val == atlas_irrationality(str)
true
source

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)
Root 0.309017 + 0.951057*im of x^4 + x^3 + x^2 + x + 1

julia> y^5
Root 1.00000 of x - 1

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 — Method
gen(K::QQAbField, s::String)

Return the generator of the abelian closure K that can be used to construct primitive roots of unity. The string s will be used during printing.

source
set_variable! — Method
set_variable!(K::QQAbField, s::String)

Change the printing of the primitive n-th root of the abelian closure of the rationals to s(n), where s is the supplied string.

source
get_variable — Method
get_variable(K::QQAbField)

Return the string used to print the primitive n-th root of the abelian closure of the rationals.

source

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

Return the element of F that is the $p$-modular reduction of val, where $p$ is the characteristic of F. An exception is thrown if val cannot be reduced modulo $p$ or if the reduction does not lie in F.

Examples

julia> K, z = abelian_closure(QQ);

julia> F = GF(2, 3);

julia> reduce(z(7), F)
o
source