Support basic Ideal properties

[wardjm]

Suggestion: copy what Sage does to answer questions about basic Ideal properties.

R.<z> = PolynomialRing(FiniteField(7))
P = Ideal(R, z^4-1,z^3-3*z^2+3*z-1)
P.is_principle()
True

Sage has

• is_principle
• is_prime
• is_primary
• is_maximal
• is_idempotent
• is_reducible? -- I don't see this one in tab-completion, but it could be nice to have

Knowing these properties can likely speed up other algorithms as well.

[fingolfin]

Hi, thanks for your interest!

This package is mostly focus on providing some general interfaces for algebraic structures, plus some toy implementations of them primarily for testing purposes.

There isn't even a general ideal type in this package (just one for euclidean domains, so necessarily all of those would be principal ideals). To implement functions like is_prime would require adding a lot of functionality that doesn't fit into this interface package. The best this can therefore could do is add stubs for these functions (although I am not sure what is_reducible

Instead, perhaps have a look at Oscar.jl which has

• is_prime and is_primary for ideals in multivariate rings
• is_irreducible (the more common negation of your proposed is_reducible) for polynomials
  • but not for ideals there, I am not sure how hard it would be to implement; do you have a concrete use case?
• no is_principle but probably mostly because nobody needed it, should be easy to add
• no is_maximal -- IIRC this can be tricky in general but perhaps could be added, do you have a concrete use case?
• I don't know what an idempotent ideal is: perhaps one where I^2=I holds? Then of course that can be tested like that; I don't know if there are better algorithms for that. Someone could investigate if there is (you guessed it) a concrete use case ;-)

Some examples:

julia> R, (z,) = polynomial_ring(QQ,[:z])
(Multivariate polynomial ring in 1 variable over QQ, QQMPolyRingElem[z])

julia> I = ideal(R, [z^4-1,z^3-3*z^2+3*z-1])
Ideal generated by
  z^4 - 1
  z^3 - 3*z^2 + 3*z - 1

julia> is_prime(I)
true

julia> is_primary(I)
true

julia> is_irreducible(z^2)
false

julia> is_irreducible(z^2+1)
true

julia> I^2 == I
false

This is all with a multivariate polynomial ring in one variable. Annoyingly not much seems to be there yet for the true univariate case. It shouldn't be hard to add that (to Hecke, I guess, @thofma ? should I ask one of our students to work on it?)

julia> R, z = QQ[:z]
(Univariate polynomial ring in z over QQ, z)

julia> I = ideal(R, [z^4-1,z^3-3*z^2+3*z-1])
ideal(z - 1)

julia> is_prime(I)
ERROR: MethodError: no method matching is_prime(::Hecke.PIDIdeal{QQPolyRingElem})
The function `is_prime` exists, but no method is defined for this combination of argument types.

julia> I^2  # oops
ERROR: MethodError: no method matching ^(::Hecke.PIDIdeal{QQPolyRingElem}, ::Int64)
The function `^` exists, but no method is defined for this combination of argument types.

julia> I*I == I
false

I've submitted a patch with the missing ^ in thofma/Hecke.jl#1813 for the other methods it'll take a bit more time.

[fingolfin]

Some more partial progress in thofma/Hecke.jl#1825

In general this issue here however is part of the older issue #1733 so I am going to close this here and add some more details there.