Add Hilbert class fields

[jwj61]

Where possible, compute/store/display a polynomial for the Hilbert class field of number fields.

[AndrewVSutherland]

+1 to this. It would also be nice to know when a number field is the Hilbert class field of another number field in the database.

[jwj61]

That should not be hard provided we make an index for that entry.

Since it would be stored as a list of coefficients, which would be better, numeric[] or as a string?

[AndrewVSutherland]

I think the conclusion is that numeric[] is better (but @edgarcosta will correct me if I am wrong).

[edgarcosta]

I also think numeric is better, as I don't see any advantage of having the string format.

On Thu, 5 Sep 2019 at 16:45, Andrew Sutherland notifications@github.com wrote:

I think the conclusion is that numeric[] is better (but @edgarcosta https://github.com/edgarcosta will correct me if I am wrong).

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[jwj61]

Another question: should we give a polynomial which defines the Hilbert class field, or one whose compositum with the given field produces the Hilbert class field. I think the former is more natural, but the latter is what pari produces and leads to smaller degree polynomials.

[AndrewVSutherland]

I would certainly prefer the former (all I would do with the later is take the compositum and wind up with a polynomial that I then want to polredbest)

[JohnCremona]

This is surely just a relative polynomial vs. absolute polynomial question for a fields which is most naturally presented as a relative extension. I think that something would be lost if the Hilbert classfield of Q(sqrt(-23)) did not mention the polynomial x^3-x^2+1 of discriminant -23. So, perhaps both?

[jwj61]

Personally, I would lean toward absolute polynomials, but am amenable.

To the extent I have seen with quadratic fields, pari always gives a relative polynomial Q[x]. Since it affects how the database entries are set up, is there a theorem that the relative polynomial can always be taken with coefficients in the base field?