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 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 -- 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
