Bill Allombert on Sat, 24 Aug 2019 10:27:11 +0200

 Re: the minimal polynomial over the composite field

```On Sat, Aug 24, 2019 at 01:11:38PM +0900, macsyma wrote:
> > Bill Allombert on Fri, 23 Aug 2019 12:17:19 +0200
> > Could you give more background about why you would like to compute that ?
> My goal is to create a procedure based on Galois theory
> that expresses all roots of given polynomial by radicals
> (mentioned earlier K is the base field of the tower of radical extensions),
> and I have already written a code using PARI/GP,
> GAP (for getting the composition series),
> Nemo/Julia and Maxima (for the calculations with algebraic numbers
> defined by multivariate polynomials).
> It has performance comparable to MAGMA's SolveByRadicals()
> and it don't make a composite reduction like GAP's RootsOfPolynomialAsRadicals().
> However, I posted some parts of processing in search of room for improvement.
>
> > In particular, are your polynomials g random ?
> > (in your examples they ave very specific shapes which allow to do the
> > computation nearly by hand).
> The reason for selecting trivial (in a sense) x^n-2 as an example is
> simply the shape is concise. The input polynomial for my procedure is
> in principle arbitrary as long as it is solvable and irreducibele over Q.
> For example, if we input a polynomial in your database
> http://pari.math.u-bordeaux.fr/galpol/30/1/index.html ,
> it outputs
> http://macsyma.starfree.jp/maxima/galpol30.txt ,
> where A is a primitive element of a splitting field of the input,
> [p,v] = [the defining polynomial of v, a radical or a root of unity],
> since it is too long, the representation of roots by A is omitted.

Do you assume g to be Galois ? It seems to me you should be able to know
whether the polynomial will factor or not by computing conductor of
cyclic subfields.

Cheers,
Bill

```