Kurt Foster on Thu, 25 Mar 2010 17:52:49 +0100

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Low-degree test polynomials with different signatures

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.

I'm aware of the GALPOL database of defining polynomials for normal extensions of Q with Galois groups G of order up to 110. For those G which are weakly supersolvable, of course, I could probably use Pari's galoissubfields() function to get some examples with different signatures in some cases.

The particular case that got my attention was the degree-6 group 6T14 = PGL(2,5) of order 120, just beyond the range of the GALPOL database. It's not hard to show that a degree-6 polynomial in Q[x] with this Galois group can *not* have signature [4,1], but the test polynomial in the above-mentioned list has signature [0,3]. I assume signature [6,0] is possible, and have been too lazy to consider whether signature [2,2] is possible -- yet.

Of course for groups G of order n whose prime-order subgroups are all normal, irreducible polynomials in Q[x] with Galois group G must have degree n, so the zeroes must be either all real or all complex.

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?

If there are no such tables, are there relatively simpleminded ways of constructing examples?