Loïc Grenié on Wed, 18 Jun 2014 10:14:33 +0200 |
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Re: Creating Q(a1,a2,...,an) |
2014-06-17 23:59 GMT+02:00 Zickert, Christian <zickert@math.umd.edu>: > Hello, > > Is there a way in Pari to create the number field generated by elements in a > fixed number field? > > More precisely, suppose p in Q[x] is irreducible, and that q_1,...,q_n are > polynomials in Q[x]. > One can then create the elements a_i=Mod(q_i(x),p) in the number field > defined by p. K=nfinit(p); a_i=-polcoeff(nffactor(K,q_i)[1,1],0) This is not safe as it does not check whether q_i effectively factors in K. It works if p has a variable of lower order than q_i. In practice it usually means p has variable y and the q_i's have variable x. You can use subst(p,'x,'y) in case p has variable x. To have a number field in which q_1, q_2, q_3 have (at least) a simple root, you can use polcompositum: p=q_1; p=polcompositum(p,q_2);p=p[#p] p=polcompositum(p,q_3);p=p[#p] p=subst(p,'x,'y); > I would like a command that takes p, and the q_i's as input and outputs a > polynomial q generating the number field Q(a_1,...,a_n), as well as > expressions of the a_i's in terms of q. > > Surely, someone must have written a command that does that. > > If someone could provide a script, or some hints for how to obtain this, I > would really appreciate it. Left as an exercise for the reader. Loïc