Add "Browse by families" to number fields homepage

[rvisser7]

I think it would be very nice to have a "Browse by families" row near the top of the number fields homepage, as is already done in the abstract groups and modular curves sections.

Here are some examples of families which, at least for me, would be very useful to include (at present, there doesn't seem to be an easy way to access search pages for these via the existing search interface):

• Cyclotomic fields $\mathbb{Q}(\zeta_n)$.
• Maximal real subfields of cyclotomic fields, i.e. $\mathbb{Q}(\zeta_n + 1/\zeta_n) = \mathbb{Q}(\zeta_n)^+$.
• Pure cubic fields $\mathbb{Q}(\sqrt[3]{n})$, or more generally all pure fields $\mathbb{Q}(\sqrt[m]{n})$.
• Shanks's family of simplest cubic fields, i.e. the cubic fields with defining polynomial $x^3 - ax^2 - (a+3)x - 1$, for some $a \in \mathbb{Z}$.
• Ennola's family of cubic fields, i.e. the cubic fields with defining polynomial $x^3 + (a-1)x^2 - ax - 1$ for some $a \in \mathbb{Z}$.

(I'm sure others will have other examples of families to suggest!)

Another related proposal could also be to add a "Browse by property" row, i.e. having links to fields which are abelian, dihedral, solvable, CM, Galois, monogenic, etc. (this is similarly done in the abstract groups, Galois groups, and abelian varieties sections). Of course one can already easily filter by these properties in the search boxes, so this would simply be a convenience feature. Or another option might be to merge the families and properties row into a single "Browse" row. :)

[havarddj]

This is a great idea! Perhaps one should simply make a new database with families? Additionally, we could record membership in each number field row, so we can include a link to relevant families in the "related objects" section of the property box (so we don't have to look it up).

[jwj61]

Browsing by property sounded interesting at first, but then I thought of the results. A search for abelian, solvable, Galois or monogenic fields will produce $$Q$$ followed by quadratic fields as far as the eye can see. CM is similar, but only showing quadratic imaginary fields. Since this section is for complete novices, that could even be considered a problem since it may give them the wrong impression.

The idea of a family given by polynomial with parameters has come up before. This has some connection to Belyi maps, but most of those have a different monodromy group over $$\mathbf{Q}$$ compared to $$C$$, and you may not get particularly interesting examples. Still it would be interesting to have a database of parameterized fields (which could include bullet points 3, 4, and 5 above). One needs to collect a substantial number of these and decide on specialization points, and also identify existing fields which might fit the family even though our defining polynomial may be of a different form. For what I envision, there may be a different search page for families as well as home pages for each family.

Full cyclotomic fields and their maximal real subfields is another type of family. They should be easy to implement since one can precompute full list (which may already be in webnumberfield.py).

[jwj61]

One more comment, the family of simplest cubic fields is generic for Galois group $$C_3$$, so the list of fields can be gotten by searching for that Galois group. I think the papers may be mainly interested in connecting invariants to the parameter, but the parameter may be lost when converting to our polynomial.