Bill Allombert on Thu, 04 Dec 2003 19:42:19 +0100

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Re: Computing a system of fundamental units

On Tue, Dec 02, 2003 at 01:59:08AM -0500, McLaughlin, James wrote:
> A little more on these polynomials. The ones I am interested are of prime degree p, p >= 29. 
> (smaller degrees I am able to deal with) All are in Z[x].
> All their roots are real - is anything special known about the degree  of the splitting field of a polynomial whose roots are all real?

No, it is widely conjectured that this give no information on the Galois
group and the splitting field.

> I suspect that the degree of the splitting field is n*(n-1), but see no way to prove it and know of no results in this direction.

[Is n prime ?]

I believe there are special technique for this case, but I don't
remember the detail off-hand.

Cebotarev-van der Waerden estimates should give very good result, while
not certified in all case.

> The reason I would like some better bound d on the degree  of the splitting field than the crude bound d  = n! is that I am working with some Thue equations and want to apply the Baker-Wustholz theorem on linear forms in logarithms. The smaller the bound d, the less precision I need.
> I am not expert in using pari/gp and the only relevant command I know is "polgalois", but this only works up to degree 11 on the version I have.

Magma support degree <= 30, for what it worth.