Thanks for the replies! Over other fields it's no good just extending by a root of the n'th > cyclotomic polynomial, since that need not be irreducible! The example you > gave was particularly unfortunate since over GF(5) the 5th cyclotomic poly > has only 1 root with multiplicity 4. So it's quite right to say that the > resulting algebra is not a field. (The question about Sage not being able > to tell that the result is finite is unfortunate, and should be logged as a > feature to be implemented., but is not particularly relevant to the current > discussion). > > I clearly shouldn't post these things late at night! In the context that I am working with a "primitive pth root of unity in characteristic p" is an important special case, but this corresponds to xi=1 so I can simply work with GF(p).
For fields of characteristic p>0, I need to work in GF(p^a) for some a so I guess that my question really is: does anyone know how to construct the smallest extension of GF(p) which contains a primitive eth root of unity when gcd(e,p)=1? Andrew -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
