Loïc Grenié on Wed, 18 Jun 2014 10:14:36 +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.


   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:


> 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.