Fix for denominator_ideal function

[bleveque]

The denominator_ideal of 0 should return the ideal (1), but it currently returns a ValueError.

[bleveque]

fixes denominator_ideal for input 0

[jdemeyer]

The problem really is that Sage only supports the zero ideal and fractional ideals of number fields. Sage simply does not know about the ideal (1) in a number field.

So proposing to close this as "wontfix".

[davidloeffler]

I'm with Jeroen on this. I think ValueError? is the right behaviour here. Positive review to the proposal to close as wontfix.

[was]

Huh? Obviously, the denominator ideal of 0 should be the unit ideal, just at the denominator if 0 is 1:

sage: K.<a> = NumberField(x^2 + 1)
sage: K(0).denominator()
1

The statement "Sage only supports the zero ideal and fractional ideals of number fields. Sage simply does not know about the ideal (1) in a number field." sounds like nonsense to me. The ideal generated by 1 is a perfectly good fractional ideal. That's why people say "the *group* of fractional ideals" -- groups have an identity element.

sage: K.maximal_order().ideal(1)
Fractional ideal (1)

[was]

Also, I give this patch a positive review.

[davidloeffler]

Sage simply does not know about the ideal (1) in a number field." sounds like nonsense to me. The ideal generated by 1 is a perfectly good fractional ideal.

What I believe Jeroen meant here is not the OK-submodule OK of K, but the K-submodule K of K -- the unit ideal of K in the purely ring-theoretic sense. This is indeed an object Sage genuinely doesn't know about -- we special-case the zero ideal, but not the whole-of-K ideal.

This is just FYI; I'm not trying to dispute your decision to give the patch a positive review. Frankly I'm not convinced that the denominator of 0 is worth losing much sleep over.

[jdemeyer]

Please DO NOT reopen tickets!

If you disagree with me closing the ticket, that's fine, just state it in the comments.

But DO NOT reopen tickets!

You can totally disagree with me about closing a ticket. I am a reasonable person and will very likely listen. But it's very annoying if my idea of which tickets are open/closed does not correspond to reality.

So, DO NOT reopen tickets!

[jdemeyer]

Replying to davidloeffler:

What I believe Jeroen meant here is not the OK-submodule OK of K, but the K-submodule K of K -- the unit ideal of K in the purely ring-theoretic sense. This is indeed an object Sage genuinely doesn't know about -- we special-case the zero ideal, but not the whole-of-K ideal.

This is what I meant, yes. But it is indeed mostly matter of definition.

I don't understand why the patch changes the documentation of the function. I think the old documentation is more clear and I think it also justifies better the choice that the numerator of (0) is the fractional ideal (1).

However, if you want to define the denominator_ideal of zero to be the fractional ideal (1), then certainly the numerator_ideal of zero should be (0). So this still needs work.

[bleveque]

Hello,

In response to Jeroen's comment, I opened a new ticket --  http://trac.sagemath.org/sage_trac/ticket/12046 -- with a proposed change to the numerator_ideal function. Sorry if it's confusing having two tickets dealing with a similar issue.

-Ben