Bill Allombert on Thu, 25 Mar 2010 19:30:49 +0100

 Re: Low-degree test polynomials with different signatures

```On Thu, Mar 25, 2010 at 10:51:50AM -0600, Kurt Foster wrote:
> The usual lists of "test polynomials" for transitive groups of given
> degree n (for example
> http://world.std.com/~jmccarro/math/GaloisGroups/GaloisGroupPolynomials.html
>  for degrees up to 9) usually give only one polynomial of degree n per
> group.  However, there is usually more than one possibility for the
> signature.  I was unable to find a list giving polynomials with each
> possible signature per group.

> So I was wondering: Are there tables of all possible signatures for
> irreducible polynomials of degree n having a given transitive group G of
> degree n as Galois group for n up to 8 or 10 or something?  And if so,
> are there corresponding test polynomials for each case?

I think you are looking for Klueners-Malle database.
<http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html>

For example for 6T14:

Sig = 0: x^6 + 3*x^4 - 2*x^3 + 6*x^2 + 1
Sig = 2: x^6 - 2*x^5 - x^4 + 4*x^3 - 4*x^2 + 4*x + 1
Sig = 6: x^6 - 18*x^4 + 9*x^3 + 90*x^2 - 70*x - 69

Cheers,
Bill

```