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