galthy.jl / van_hoeij: must never be zero

[benlorenz]

Describe the bug
Rare error must never be zero in test/NumberTheory/galthy.jl:109 in our CI, e.g.:
https://github.com/oscar-system/Oscar.jl/actions/runs/10960677682/job/30435848700#step:9:7123.
The issue might be in fixed_field or somewhere in Hecke?

To Reproduce
After trying some seeds I found that this code reproduces the error (on three different linux computers with julia 1.10.5 and an M1 mac with julia 1.10.4):

julia> R, s = QQ[:s];
julia> K, q = number_field(s^2 - 2, "q");
julia> Kw, w = polynomial_ring(K, :w);
julia> f = w^16 - 32*w^14 - 192*w^12 + 22720*w^10 + 23104*w^8 - 2580480*w^6 + 41287680*w^4 + 106168320*w^2 + 84934656;
julia> g, s = galois_group(f);
julia> ss = map(representative, subgroup_classes(g));
julia> H = ss[1000];

julia> Oscar.randseed!(15290);

julia> f = fixed_field(s, H)
ERROR: must never be zero
Stacktrace:
  [1] error(s::String)
    @ Base ./error.jl:35
  [2] van_hoeij(f::AbstractAlgebra.Generic.Poly{AbsSimpleNumFieldElem}, P::AbsSimpleNumFieldOrderIdeal; prec_scale::Int64)
    @ Hecke ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/PolyFact.jl:759
  [3] van_hoeij(f::AbstractAlgebra.Generic.Poly{AbsSimpleNumFieldElem}, P::AbsSimpleNumFieldOrderIdeal)
    @ Hecke ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/PolyFact.jl:541
  [4] factor_new(f::AbstractAlgebra.Generic.Poly{AbsSimpleNumFieldElem})
    @ Hecke ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/PolyFact.jl:307
  [5] _factor(f::AbstractAlgebra.Generic.Poly{AbsSimpleNumFieldElem}; algo::Symbol)
    @ Hecke ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/Elem.jl:354
  [6] _factor
    @ ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/Elem.jl:346 [inlined]
  [7] is_irreducible
    @ ~/.julia/packages/Hecke/7wI0D/src/NumField/NfAbs/Elem.jl:418 [inlined]
  [8] number_field(f::AbstractAlgebra.Generic.Poly{AbsSimpleNumFieldElem}, S::String; cached::Bool, check::Bool)
    @ Hecke ~/.julia/packages/Hecke/7wI0D/src/NumField/NfRel/NfRel.jl:187
  [9] number_field
    @ ~/.julia/packages/Hecke/7wI0D/src/NumField/NfRel/NfRel.jl:185 [inlined]
 [10] #extension_field#305
    @ ~/software/polymake/julia/Oscar.jl/src/NumberTheory/GaloisGrp/GaloisGrp.jl:2220 [inlined]
 [11] extension_field
    @ ~/software/polymake/julia/Oscar.jl/src/NumberTheory/GaloisGrp/GaloisGrp.jl:2219 [inlined]
 [12] fixed_field(GC::Oscar.GaloisGrp.GaloisCtx{Hecke.vanHoeijCtx}, U::PermGroup, extra::Int64)
    @ Oscar.GaloisGrp ~/software/polymake/julia/Oscar.jl/src/NumberTheory/GaloisGrp/GaloisGrp.jl:2374
 [13] fixed_field(GC::Oscar.GaloisGrp.GaloisCtx{Hecke.vanHoeijCtx}, U::PermGroup)
    @ Oscar.GaloisGrp ~/software/polymake/julia/Oscar.jl/src/NumberTheory/GaloisGrp/GaloisGrp.jl:2339
 [14] top-level scope
    @ REPL[10]:1

When setting a different seed the code runs without errors (usually).

Expected behavior
No error.

System (please complete the following information):

julia> Oscar.versioninfo(full=true)
OSCAR version 1.2.0-DEV - #master, 2fe164e918 -- 2024-09-30 18:04:55 +0200
  combining:
    AbstractAlgebra.jl   v0.43.2
    GAP.jl               v0.12.0
    Hecke.jl             v0.34.4
    Nemo.jl              v0.47.1
    Polymake.jl          v0.11.21
    Singular.jl          v0.23.8
  building on:
    FLINT_jll               v300.100.300+0
    GAP_jll                 v400.1300.102+2
    Singular_jll            v404.0.606+0
    libpolymake_julia_jll   v0.12.1+0
    libsingular_julia_jll   v0.45.5+0
    polymake_jll            v400.1200.1+0
See `]st -m` for a full list of dependencies.

Julia Version 1.10.5
Commit 6f3fdf7b362 (2024-08-27 14:19 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 8 × 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, tigerlake)
Threads: 1 default, 0 interactive, 1 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_PRECOMPILE_TASKS = 3
Official https://julialang.org/ release