is_prime buggy for multivariate polynomial rings

[simonbrandhorst]

By definition the (1) ideal is never prime.
Yet:

julia> R,x = polynomial_ring(QQ,2);

julia> I = ideal(R,[1])
ideal(1)

julia> is_prime(I)
true

The reason is that

julia> minimal_primes(I)
1-element Vector{MPolyIdeal{QQMPolyRingElem}}:
 ideal(1)

Refuses to return an empty list but instead returns a non-prime ideal.
(Matching its documentation, so it is not a bug.)
In any case I would suggest to rather return only prime ideals,
even if it means that you cannot do
intersect(minimal_primes(I)) (if the return value is the empty array). One can still do something like
intersect(minimal_primes(I), init=R) instead.
We have the same situation when we sum the elements of an array.
@wdecker

[fingolfin]

We discussed this today and agreed that minimal_primes should return an empty vector in this case. Moreover, there are other functions that may suffer from similar problems, so we should check those as well. E.g. this is also questionable:

julia> primary_decomposition(I)
1-element Vector{Tuple{MPolyIdeal{QQMPolyRingElem}, MPolyIdeal{QQMPolyRingElem}}}:
 (ideal(1), ideal(1))

There are further considerations (e.g. maybe primary_decomposition should return a special type; or should always include a "unit factor" etc.) but that goes beyond the point here.

[fingolfin]

@wdecker agreed to work on this. Meaning: just changing minimal_primes as discussed, verifying is_prime now works, perhaps adding a test. And depending on his time also look for other problematic commands that need similar fixes.

[wdecker]

Solved with PR #2473

[simonbrandhorst]

Thank you!