Bill Allombert on Tue, 02 Dec 2003 00:56:16 +0100

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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.