On Thu, Oct 13, 2016 at 5:09 PM, Kwankyu Lee <ekwan...@gmail.com> wrote:

> Hi David,
> First, thank you for technical explanations. They are compelling.
> On Thursday, October 13, 2016 at 9:59:50 PM UTC+2, David Roe wrote:
>> In order to create finite fields with arbitrary variable names that fit
>> into a lattice of fields, one possibility would be able to give an
>> algebraic closure explicitly as an argument to GF.  Is that what you're
>> suggesting?
>> sage: F = GF(3).algebraic_closure('custom')
>> sage: K9 = GF(9,'b',Fpbar=F)
>> sage: K81 = GF(81,'c',Fpbar=F)
>> sage: K81.has_coerce_map_from(K9)
>> True
> Still speaking from a user's point of view, is there also a technical
> difficulty in keeping just one lattice of fields? So a new creation of a
> finite field gets automatically fitted into the canonical lattice?

You want to be able to create finite fields that don't fit into the
lattice, because as the degree increases it becomes very expensive to
compute polynomials that are compatible with the other extensions that
you've already constructed.  In contrast, constructing a finite field using
a random polynomial is relatively cheap.  Currently, Sage uses the presence
or absence of a specified variable name to determine whether you want a
field that fits into this lattice or not.

One application for having multiple lattices is to support algebraic
closures for non-prime fields (which is not yet implemented in Sage).

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