Hi there, I have given some thought to finite extension field isomorphisms and finite extension field embeddings.
As I realized that my number theory skills are quite weak I need some input. Let K1.<a> = GF(q) and K2.<b> =GF(q) with possible distinct minimal polynomials. I understand now how to construct an isomorphism from K1 to K2 (lets call it phi) and vice versa (phi^(-1)) and I have put some code at https://sage.math.washington.edu:8102/home/pub/14/ There are however some issues left: * I think having a == phi^(-1)(phi(a)) is a desired property. Is there any smarter way of constructing phi^(-1) from phi than checking all homomorphisms from K2 to K1 if they fullfill the equality? * 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? Martin -- name: Martin Albrecht _pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99 _www: http://www.informatik.uni-bremen.de/~malb _jab: [EMAIL PROTECTED] --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
