Sage does have a function is_norm() for number field elements so the underlying algebraic problem should be solvable.
Example (p=3): sage: Q3.<z> = CyclotomicField(3) sage: a=2+3*z sage: b=3+4*z sage: x=polygen(Q3) sage: L.<a3>=Q3.extension(x^3-a) sage: b.is_norm(L) False On 10 July 2017 at 14:23, Pierre <pierre.guil...@gmail.com> wrote: > Hi all ! > > I wanted to know whether Sagemath had any support for cyclic algebras. From > the manual, I strongly suspect the answer is "no", but you never know. > > Let me be more concrete. For a prime p, let K be QQ with the p-th roots of > unity adjoined. For a, b in K, there is a cyclic algebra (a,b) over K > (technically, this depends on a choice of primitive root in K); for p=2, > this is the quaternion algebra (a, b) over QQ. > > I would like to be able to answer questions such as: is (a,b) trivial? For > p=2, Sage does this, essentially with hilbert_conductor(a,b). > > Also, (a,b) is trivial if and only if b is a norm from K[a^(1/p)]. Finding > explicitly an element from this field whose norm is b would be awesome. When > p=2 and so K=QQ, it's a matter of finding x, y in QQ such that x^2 - ay^2 = > b, and trying random values for x and y (essentially...) works fine. Over > more complicated fields, PARI has functions accessible through sage to find > points on conics. > > If any of the above can be facilitated by Sage for p odd, it would be great. > > Thanks! > Pierre > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
