going from a complex field to a real field, and vice-versa

[laurentbartholdi]

I'm trying to get hold of the real field associated with a complex field:

julia> ℂ = AcbField()
Complex Field with 256 bits of precision and error bounds

julia> real_field(ℂ)
ERROR: UndefVarError: `real_number_field` not defined in `Hecke`
Suggestion: check for spelling errors or missing imports.
Stacktrace:
 [1] real_field(args::AcbField; kwargs::@Kwargs{})
   @ Hecke ./deprecated.jl:116
 [2] real_field(args::AcbField)
   @ Hecke ./deprecated.jl:113
 [3] top-level scope
   @ REPL[120]:1

julia> ℝ = parent(real(one(ℂ)))
Real Field with 256 bits of precision and error bounds

There must be a simpler solution...

EDITOR'S NOTE: the following is copied from #2097

Given an ArbField, how do I construct the associated complex field?

julia> ℝ = ArbField(666)
Real Field with 666 bits of precision and error bounds

julia> ℂ = complex(ℝ) # BOOM
ERROR: MethodError: no method matching complex(::ArbField)
The function `complex` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  complex(::Type{Missing})
   @ Base missing.jl:111
  complex(::Missing)
   @ Base missing.jl:101
  complex(::Type{T}) where T<:Real
   @ Base complex.jl:195
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[140]:1

julia> complex(Float64) # works
ComplexF64 (alias for Complex{Float64})

julia> ℂ = AcbField(precision(ℝ)) # ugly and non-generic
Complex Field with 666 bits of precision and error bounds

[fingolfin]

There is currently no concept of a "real field associated to a complex field" (or conversely). Probably main because nobody had a need for it.

I think a patch to add functions doing that would have a good chance of being merged. I don't know what good names would be... If we do real_field and complex_field then user might also expect to be able to call them with no arguments, acting aliases for RealField() and ComplexField. That might be nice to have anyway.

[fingolfin]

I guess we could add something like

real_field(F::AcbField) = ArbField(precision(F))
real_field(F::ComplexField) = RealField()

Alternatively we could have a constructor

ArbField(F::AcbField) = ArbField(precision(F))

and then dually for the other direction...

[fingolfin]

Of course writing ArbField(precision(F)) is also not that bad...