Can we make ComplexField(QQ) ?

[zhaoli-IHEP]

Is it possible to have the rational complex field instead of the float number complex field?
Meanwhile maybe fit into LaurentSeriesField(ComplexField(QQ), 5, "ep")?

[thofma]

It is already there, but maybe not under the name that you would have in mind.

julia> Qx, x = QQ["x"]
(Univariate Polynomial Ring in x over Rational Field, x)

julia> CC, i = NumberField(x^2 + 1, "i")
(Number field over Rational Field with defining polynomial x^2 + 1, i)

julia> i^2
-1

(https://en.wikipedia.org/wiki/Gaussian_rational)

Do you have a suggestion for an alias? Would you suggest ComplexField(QQ)?

[zhaoli-IHEP]

It is already there, but maybe not under the name that you would have in mind.

julia> Qx, x = QQ["x"]
(Univariate Polynomial Ring in x over Rational Field, x)

julia> CC, i = NumberField(x^2 + 1, "i")
(Number field over Rational Field with defining polynomial x^2 + 1, i)

julia> i^2
-1

(https://en.wikipedia.org/wiki/Gaussian_rational)

Do you have a suggestion for an alias? Would you suggest ComplexField(QQ)?

So great! Thank you.
I think ComplexField(QQ) would be nice.

[thofma]

Let's hear what the others think. @wbhart @tthsqe12

[wbhart]

I don't have any objections. It would be nice if it also worked for Nemo.

[thofma]

This is a about Nemo.

[tthsqe12]

I have a reservation about ComplexField(QQ): it suggests that we can do ComplexField(F) for a multitude of fields F, ie extent F by a nonreal something. What is ComplexField(GF(5)) or ComplexField(GF(7))?
If you share my reservations, name them GaussianRationals() and GaussianIntegers(). Otherwise, call it whatever you want. Just make sure all instances hash equal!

On a related note, Complex{fmpq} doesn't work :(

[thofma]

After sleeping one night over this, I also don't like it too much anymore. It does not really play nice with im and should it return both the field and the imaginary unit? It is also awkward because one needs to use i = gen(K) to get the imaginary unit.

[tthsqe12]

So, I have come across the need for exactly this kind of thing. There are parents ZZi and QQi with elem types fmpzi and fmpqi. The behaviour is the following (I am not willing to compromise on the denominator(x) != 25 thing).

julia> i = ZZi(im)
im

julia> typeof(ans)
fmpzi

julia> x = (3+4i)//(5+10i)
(11 - 2*im)/25

julia> typeof(ans)
fmpqi

julia> (numerator(x), denominator(x))
(2 + im, 4 + 3*im)

julia> map(factor, ans)
(1 * (2 + im), im * (2 - im)^2)

It is supposed to play nice with the other complex types:

julia> CC = AcbField(50); gamma(CC(1/2,1/3))
[1.1789716859081 +/- 3.88e-14] + [-0.7570793934981 +/- 4.63e-14]*im

julia> simplest_rational_inside(ans)
(38171685330064 - 24512035975001*im)/32377100982424

(This may or may not be the simplest rational inside, it is just for demo purposes)

[fingolfin]

We won't add ComplexField(QQ) (or complex_field(QQ)) but indeed QQi exists in Nemo, and quadratic_field in Hecke / Oscar