The GF function (see also #1906)

[JohnAAbbott]

I would like to make some comments, suggestions, questions about the GF function.

Everything pertains to using Oscar; #--> v 0.11.2 (2023-02-09)

1. Error message

julia> GF(1)   #ERROR: DomainError with 1: Characteristic must be prime
julia> GF(0)   #ERROR: DomainError with 0: Characteristic must be positive
julia> GF(-1)  #ERROR: DomainError with -3: Characteristic must be positive

Is a single error message more helpful?
Combined error message? "Characteristic must be positive & prime"
Since the error message includes the troublesome value, it is not necessary to say which criterion failed.

2. Possible signatures of GF

I wonder how "Closest candidates" chooses the methods it displays?

julia> GF(3)
julia> GF(BigInt(3)) # ERROR: MethodError: no method matching GF(::BigInt)
Closest candidates are:
  GF(::Integer, ::Int64)
  GF(::Integer, ::Int64, ::Union{AbstractString, Symbol}; cached)
  GF(::Int64; cached)

However, asking explicitly for a list of signatures gives more choices:

help?> GF
search: get_field gfp_fmpz_mat gfp_fmpz_poly gfp_fmpz_elem gfp_fmpz_mpoly

  No documentation found.

  Nemo.GF is a Function.

  # 7 methods for generic function "GF":
  [1] GF(n::Int64; cached) in Nemo at /home/zez25zuh/.julia/packages/Nemo/g02iz/src/flint/gfp_elem.jl:471
  [2] GF(n::UInt64; cached) in Nemo at /home/zez25zuh/.julia/packages/Nemo/g02iz/src/flint/gfp_elem.jl:478
  [3] GF(p::Integer, k::Int64) in Hecke at /home/zez25zuh/.julia/packages/Hecke/TlFaj/src/Misc/FiniteField.jl:15
  [4] GF(p::Integer, k::Int64, s::Union{AbstractString, Symbol}; cached) in Hecke at /home/zez25zuh/.julia/packages/Hecke/TlFaj/src/Misc/FiniteField.jl:15
  [5] GF(n::fmpz; cached) in Nemo at /home/zez25zuh/.julia/packages/Nemo/g02iz/src/flint/gfp_fmpz_elem.jl:458
  [6] GF(p::fmpz, k::Int64) in Hecke at /home/zez25zuh/.julia/packages/Hecke/TlFaj/src/Misc/FiniteField.jl:16
  [7] GF(p::fmpz, k::Int64, s::Union{AbstractString, Symbol}; cached) in Hecke at /home/zez25zuh/.julia/packages/Hecke/TlFaj/src/Misc/FiniteField.jl:16

3. 2-arg version is fragile

Here are several examples of "sub-optimal behaviour". I hope some of them can be added to a test suite!

GF(0,0)  # --> Finite field of degree 0 over F_0
GF(0,1)  # ERROR: DivideError: integer division error
GF(0,2)  # ERROR: DivideError: integer division error

GF(1,0)  # Finite field of degree 0 over F_1
GF(1,1)  # Finite field of degree 1 over F_1
GF(1,2)  # infinite loop?
## [1] (::Nemo.var"#39#44"{fmpz, Int64, Symbol})()
##   @ Nemo ~/.julia/packages/Nemo/g02iz/src/flint/FlintTypes.jl:2146

GF(1,2^33)
## Exception (FLINT memory_manager). Unable to allocate memory (68719476744).
## ERROR: Problem in the Flint-Subsystem

GF(1,2^62)  # SEGV

GF(1,2^61)
##  signal (11): Segmentation fault
##  in expression starting at REPL[2]:1
##  malloc(): unaligned fastbin chunk detected
##
##  signal (6): Aborted
##  in expression starting at REPL[2]:1
[[julia is now in a sort of "zombie" state]]


### -- More meaningless inputs (for 2-arg version):
GF(0,-1)  # SEGV
GF(4,-1)  # SEGV
GF(-1,-1) # SEGV

[fingolfin]

See #867. I think the work by @thofma on unified finite fields will take care of most of this. We definitely should revisit it once that work has landed.

Of course we could fix some of them right away, but I'd be careful to coordinate this with @thofma to avoid breaking his patches and thus causing even more work