Hi all, I am wondering how to make some computations with rather specific field extensions. I cannot figure out how to solve the following on SAGE.
Mathematically, I have the following field: Q(x,y,z,t,a) Where x,y,z,t are indeterminates and "a" is an algebraic number over the rationals (lets say degree 4). If I have some elements, let say f,g,h in this field I would like to: (1) Obtain a representation "in base a", that is, write f as f0 + f1*a + f2*a^2 + f3*a^3 (2) Compute the norm of f with respect to the extension [Q (x,y,z,t,a) : Q(x,y,z,t) ] (3) Extract a valid numerator and denominator of f, that is, compute polynomials r,s in Q[x,y,z,t,a] such that f=r/s (4) Construct ideals and compute a Grobner basis of a series of polynomials extracted as numerators of rational fuctions (5) Factorize polynomials in Q[x,y,z,t,a] extracted from numerators/denominatos of rational functions. Depending on how I represent the field I think that I cannot do almost all of the above, for example I have tried working with R.<x,y,z,t,a>=QQ['x,y,z,t,a'] L=R.quotient_ring(R.ideal(a^4+a+1)) Quotient ring in general does not apply to this problem because the system cannot know if this is a field. So another approach like R.<x,y,z,t,a>=QQ['x,y,z,t,a'] R.quotient_ring(R.ideal(a^4+a+1)) neither work If I do K.<a>=NumberField(x^4+x+1) R.<x,y,z,t>=K['x,y,z,t'] I do not know how to solve problems (1), (2), (5) Is there a construction that allows me to do the previous? Maybe it is possible to construct several isomorphic fields, each one having a representation that allows to solve some of the problems above and construct explicit isomorphism between them in order to pass from one representation to another. Any hint about this? Thanks! Luis --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
