I'm designing a set of exercises for my students that are studying Galois theory for the first time. I thought it would be "fun" for them to create a tower of field extensions that creates a splitting field with a Galois group that is not solvable (S_5 in this case).
So starting with the polynomial x^5 - x - 1, I am basically adding a root at a time, factoring to get a polynomial of degree one smaller. Rinse, repeat. First iteration is below. By the time I get to degree 3 the factorizations are taking about 8 hours. My question: is there a more efficient way to do this? I know about the galois_closure() command, but it seems to take a long time as well. But most importantly, I'd like the students to have the experience of seeing each new root only creating a single linear factor. So the answer (the extension has degree 5!=120) is less important than the route they take getting to it. But maybe I'm asking for too much. Again. I have an S_4 example to fall back on, which seems to finish in reasonable time, though doesn't have the non-solvable flavor I am after. Thanks, Rob sage: M.<a>=NumberField(x^5-x-1) sage: y=polygen(M) sage: (y^5-y-1).factor() (x - a) * (x^4 + a*x^3 + a^2*x^2 + a^3*x + a^4 - 1) sage: N.<b>=NumberField(y^4 + a*y^3 + a^2*y^2 + a^3*y + a^4 - 1) ... -- 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 To unsubscribe, reply using "remove me" as the subject.
