In gmane.comp.mathematics.sage.support, you wrote: > > On Feb 12, 2:40 am, Keshav Kini <keshav.k...@gmail.com> wrote: >> I guess his question is why Sage picks "a" (generic) as a generator name >> for QQ[2^(1/3)] but "sqrt2" (hard-coded) for QQ[2^(1/2)]. >> >> -Keshav >> > > Thanks for the comments, but let me explain the question a bit more > carefully. > > As the code below shows, it doesnt seem to matter what I name the > generator. > In the case of 2^(1/3) the two fields, constructed differently and of > course isomorphic, > are actually equal. > In the case of 2^(1/2) they are not equal. > > sage: F.<a> = NumberField(x^3-2) > sage: K.<b> = QQ[2^(1/3)] > sage: F == K > True > sage: F.<a> = NumberField(x^2-2) > sage: K.<b> = QQ[2^(1/2)] > sage: F == K > False
looks weird to me. I cc this to sage-nt, just in case. > > This is reversed from how I might imagine it should work. > I expect that QQ[x]/m(x) is abstractly defined, not necessarily > embedded into CC. > On the other hand QQ[ a ] for some algebraic number a is a specific > embedding. > > In the case of deg m(x) = 2 there is only one embedding into CC, so I > can see > that sage might consider QQ[2^(1/2)] and NumberField(x^2-2) to be > equal. > (although there are two embeddings!) > > For x^3-2 there is more than one embedding into CC, so I would not > expect > sage to consider the two constructions equal. > > My main motivation is that I want to illustrate the theory of number > fields with > my students, including subtleties like the difference between a field, > defined > as the quotient of a polynomial ring, and a particular embedding. > In sage there are often many ways to construct the same object. > In some cases they are pseudonyms, in others there are fine > distinctions, > which can lead to confusion. I want to make this as smooth as > possible > for my students. > > Thanks in advance for any insight offered. > > Mike Dima -- 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
