Add ZZ[i] and QQ[i]

[tthsqe12]

context: #1110, #1072
I would be nice to have a ZZ[i] and QQ[i] that embed directly (i.e. without user intervention) into the complex numbers.
If you are asking yourself if we really need this, consider the following:

1. BigInt is to fmpz as Complex{BigInt} is to fmpzi.
2. Rational{BigInt} is to fmpq as Complex{Rational{BigInt}} is to fmpqi.
3. fmpzi has a Euclidean structure that we should be able to use
4. Unlike nf_elem, the fmpzi and fmpqi embed directly into the complex numbers so that conversions and roundings between acb are possible.

If the ZZi and QQi are completely unacceptable, please let me know and I will stop. Otherwise, I will finish the rest of the interface. Also, @thofma, I don't care any more what numerator(::fmpqi) and denominator(::fmpqi) do. They can do whatever you want. And, I don't even care if this is changed:

base_ring(a::FlintQQiField) = ZZi

[codecov]

Codecov Report

Merging #1168 (491054c) into master (4096f8d) will increase coverage by 0.33%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master    #1168      +/-   ##
==========================================
+ Coverage   86.82%   87.16%   +0.33%     
==========================================
  Files          71       74       +3     
  Lines       26681    27332     +651     
==========================================
+ Hits        23166    23824     +658     
+ Misses       3515     3508       -7     

Impacted Files	Coverage Δ
src/Nemo.jl	28.40% <ø> (ø)
src/flint/fmpq.jl	93.84% <100.00%> (+0.36%)	⬆️
src/flint/fmpz.jl	89.56% <100.00%> (+1.05%)	⬆️
src/gaussiannumbers/GaussianNumberTypes.jl	100.00% <100.00%> (ø)
src/gaussiannumbers/QQi.jl	100.00% <100.00%> (ø)
src/gaussiannumbers/ZZi.jl	100.00% <100.00%> (ø)
src/calcium/ca.jl	100.00% <0.00%> (ø)
... and 3 more

---

Continue to review full report at Codecov.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update 4096f8d...491054c. Read the comment docs.

[tthsqe12]

@albinahlback, do you have ideas for the canonicalization of the gcdx coefficients in ZZ[i]?

[albinahlback]

I have a feeling that a generalized Euclidean algorithm should return the canonical coefficients. What I mean is that via these definitions, one should be able to generalize this proof for canonical Bézout coefficients.

[tthsqe12]

I was thinking less of having the canonicity defined by an algorithm and more of having it defined by a set of easy-to-check conditions, like the fmpz_xgcd_canonical_bezout docs. I already have the gcd satsifying -pi/4 < angle(gcd(,)) <= pi/4. so that the gcd is as close the positive real axis as possible.

[albinahlback]

Ah, yeah. Let me think of a definition.

[albinahlback]

I think the easiest way to deal with this is to assume that the numbers are coprime (in sense of Gaussian integers).

Let a and b be coprime Gaussian integers such that |a|^2, |b|^2 > 1. Then I am guessing that we should be able to find x and y satisfying a x + b y = 1 such that |x|^2 \le |b|^2 / 2 and |y|^2 \le |a|^2.

And if |a|^2 = 1, then x = 1 / a and y = 0 (and vice versa).

This should be canonical Bézout coefficients. However, if this holds along with that they are unique, I do not know.

[fingolfin]

Thank you for working on this. I am fine with doing this in principle. As long as there is a good way to convert between these new elements, and existing number field elements?

Also, supporting all the features we have for the isomorphic existing field types. Like, can we also factorize in polynomial rings over QQ[i]?

[tthsqe12]

Yes, I am discussing with thofma how to make it work as a number field/ring.

[thofma]

There are a bunch of deepcopy, like in conj, which I am not sure about.

According to the documentation here

The results of deepcopy and all arithmetic operations, including powering and
division can be assumed to be new objects without other references being held,
as can objects returned from constructors.

I don't think we should extend this to other random methods. But I am happy to hear your thoughts on this.

[tthsqe12]

I don't know. I was thinking of real, imag and conj as arithmetic operations. But, if they are not "on the list" then I will not copy here. Also, what counts as a constructor? It is well-known that all over the place the parent(obj) does not necessarily create a new object.

[tthsqe12]

I think is a good start. Pros:

1. diff coverage is 100%
2. there is no documentation added whatsoever

Cons:

1. It is currently slow. I will turn on the jets later.
2. Maybe it is worth documenting that the rings ZZi and QQi exist?

[thofma]

Yay! Let's get this in.

2. Maybe it is worth documenting that the rings ZZi and QQi exist?

Yes, please do this (in a separate PR).