Search for number fields on Galois / Abelian

[alexjbest]

I was surprised that I couldn't see a way to search only for Galois or only abelian number fields. This seems like a useful feature that would probably be easiest to implement by adding a couple of columns to nf_fields or nf_fields_extras as currently it looks like these properties are computed on the fly given the Galois group. Implementing this as a fancy query is probably also not too bad.

[jwj61]

You would not want to put it in extras. Part of the reason for separating it into two tables is that we search on one, but then can pull data for pages from the other.

[jwj61]

I should also say that one can search for several Galois groups by listing them in the Galois group box. If you specify say 6T2, then it only matches sextic S3 fields, not the cubic ones. So, one could enter C2,C3,C4,C5,C6,6T2 and get all of those Galois groups.

[roed314]

I've added the relevant data on beta (columns is_galois, gal_is_cyclic, gal_is_abelian and gal_is_solvable of nf_fields). Things to do:

• Add search fields to the number field page to search on these
• Push the data to prod.
• @jwj61 should update his number field scripts to include these columns when he adds new fields.

[jwj61]

I updated the prep/import files. Of course it is untested until we have some new fields to add.

[bebr117]

I'm gonna take a look at this.

[bebr117]

I noticed that in the database, Galois groups are defined and given for non-Galois extensions (as the Galois group of the Galois closure). Should searches for extensions with Abelian/solvable Galois groups include non-Galois extensions?

[alexjbest]

I would say no, normally abelian number field in books/papers refers to Galois + abelian galois group only so I think people might get confused if we break with that convention.

[roed314]

If the Galois closure L of a field K has abelian Galois group then L=K (since K corresponds to a subgroup with trivial core (intersection of all conjugates), implying that it's just the trivial group). So the question about abelian groups is moot.

I think that it's reasonable to have any quartic number field show up as having solvable Galois group, so I would say that searches for solvable extensions should include non-Galois ones.

[JohnCremona]

Nice observation! If L is Galois abelian then all subfields are also Galois so L isn't the Galois closure of any proper subfield.

[roed314]

Yes, that's probably an easier way to see it. :-)

[alexjbest]

Looks like I misread and answered a different question with my comment, sorry!

[jwj61]

The data is there, and there is currently a search for Galois fields. The question is whether we want to push this further.

One idea is to change the yes/no selector for "Is Galois" to a dropdown with options of Cyclic, Abelian, Galois, Solvable, Any. Until we have extensions of degree 60, the results of one are always included in the results of the next so one would not need to search for these things independently. Maybe the label next to the dropdown could be "Field is ...", or something like that.