McLaughlin, James on Tue, 02 Dec 2003 08:02:10 +0100

 RE: Computing a system of fundamental units

• To: "pari-users list" <pari-users@list.cr.yp.to>
• Subject: RE: Computing a system of fundamental units
• From: "McLaughlin, James" <James.McLaughlin@trincoll.edu>
• Date: Tue, 2 Dec 2003 01:59:08 -0500
• Delivery-date: Tue, 02 Dec 2003 08:02:11 +0100
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• Thread-topic: Computing a system of fundamental units

```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?

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.

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.

Jimmy Mc Laughlin.

-----Original Message-----
From: Bill Allombert [mailto:allomber@math.u-bordeaux.fr]
Sent: Mon 12/1/2003 6:47 PM
To: pari-users list
Cc:
Subject: Re: Computing a system of fundamental units

On Mon, Dec 01, 2003 at 05:04:47PM -0500, McLaughlin, James wrote:
> I meant this response to go to the list also.
>
> Here is a related question (at least related to what I am trying to do):
>
> Given an irreducible polynomial in Z{x], is there any simple way of calculating the degree of the associated splitting field over Q?
> All I need is the degree of the extension, and not any of the other invariants of the splitting field.

[What is the degree of you polynomial ?]

In general, there are no simple way that I know of.
However there are a large number of more or less `cheap' that can be
used depending on your expectation of the result.

If you polynomial is `random' of degree n, you can expect the degree of
the spliting field to be n!. If the effective degree is close to this value,
you can prove which not to much computations. If it is close to n, the
direct method can be used.

But in the worse case, it is essentially as hard as computing the Galois
group conjugacy class of the polynomial. For this task, Magma outperform
PARI vastly.

Cheers,
Bill

```