> * How to pick a canonical isomorphism? My idea is to pick the roots > that > define phi and phi^(-1) (r1, r2) treat them as polynomials over > GF(q).base_ring() and choose the canonical isomorphism where the > product > r1*r2 is minimal. Does this make sense? > > * From there it should be easy to define finite field embeddings by > coercing > the small field to the GF(p^d) with the conway polynomial, coercing > the large > field to the GF(p^n) with the conway polynomial, and embed using > conway > polynomials. Does this make sense? >
David Roe and I started looking into this at SD3. What David Harvey said in his reply is right -- there's no way to choose a canonical map in the mathematical sense, but you can make a "default" choice. I know David Roe has thought more about this, especially wrt the coerce conversations that went on during SD4. I wasn't there for the end of that, so he should really chime in. Also, though, on the note of Conway polynomials ... it seems those are fairly hard to compute. There are these large tables of them, but they only go so far (the tables in SAGE, for instance, stop somewhere around p=65000 at n=2, 3, 4, but get smaller after that, and the largest n they hit is around 100, for p=2). While we haven't (at least as of SD3, David Roe may have by now) sat down and tried to compute out further, we have good reasons to believe it's not trivial, i.e. you wouldn't want to do it at runtime. MAGMA, for instance, doesn't do it. MAGMA is an interesting case -- they have some default (and in particular, deterministic) way of choosing polynomials to generate finite fields, but we don't know what they do. Again, I suspect David Roe has thought more about this since SD3, so he should really chime in. -cc --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
