Confusing factor results

[ThomasBreuer]

Consider the following session.

julia> using Hecke

Welcome to 

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Version 0.18.13 ... 
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julia> R, x = polynomial_ring(QQ, "x");

julia> F, z = number_field(x^2+1);

julia> f = polynomial(F, [1, 2, 3])
3*x^2 + 2*x + 1

julia> facts = factor(f)
3 * (3*x^2 + 2*x + 1)

julia> factor_squarefree(f)
1 * (x^2 + 2//3*x + 1//3)

The result of factor looks wrong, I had expected that the product of facts.unit and the a^e for (a, e) in facts yields f.

(The result of factor_squarefree could be correct if the documentation of the function would state that f is silently divided by its leading coefficient; anyhow, changing factor_squarefree such that the leading coefficient appears in the result does not fix the factor problem.)

Looking into the code, one finds that there are alternatives for computing the factors.

julia> Hecke.factor_trager(f)
1-element Vector{AbstractAlgebra.Generic.Poly{nf_elem}}:
 x^2 + 2//3*x + 1//3

julia> Hecke.factor_new(f)
1-element Vector{AbstractAlgebra.Generic.Poly{nf_elem}}:
 3*x^2 + 2*x + 1

Perhaps the functions expect that the argument is monic ...

julia> g = divexact(f, leading_coefficient(f))
x^2 + 2//3*x + 1//3

julia> Hecke.factor_trager(g)
1-element Vector{AbstractAlgebra.Generic.Poly{nf_elem}}:
 x^2 + 2//3*x + 1//3

julia> Hecke.factor_new(g)
1-element Vector{AbstractAlgebra.Generic.Poly{nf_elem}}:
 3*x^2 + 2*x + 1

... no, apparently not.
I am lost: Which of these results are intentional, where is the bug?

[ThomasBreuer]

Thanks @thofma