[GAP Forum] galois group

[MCKAY john]

In my view it is unacceptable for a result to be given by computer
unless it is true. The nature of mathematics is that it is about
certainty.

Why may a result not fit these criteria? Galois computations may
be time consuming. It is often faster to compute with numerical
approximations than to work exactly. By themselves, these
approximations are inadequate and may lead to dubious results.
A good example of how to PROVE exactness from numerical approximations
comes from a note by Darmon and Ford

(Communications in Algebra,17, (1989) pp. 2941-2943 Computational
verification of M11 & M12 as Galois groups over Q.)

referred to at the start of the issue of J. Symb. Computing devoted
to computing Galois groups. An illuminating toy example is the cubic
resolvent of a 4-ic monic polynomial in which one needs to decide
for some root labelling if x1x2+x3x4 is a rational integer. How to prove
this from numerical approximations? A priori it is an algebraic integer.

p-adic methods will often provide the means to prove exactness.
Unfortunately some systems do not give the user essential information
needed to establish whether the method is exact - and its truth assured...
as an example, testing primality is often done with many quadratic residue
tests and n tests imply a probability of 1-1/2**n of correctness.

Finally all programs have a possibility of internal error - and there is
always a remote possibility of error elsewhere (do you recall the
"division error" in a chip years ago?)

If in doubt, examine the steps of the "proof" given by GAP.

GAP is one of the very best and, as far as  know, GAP provides
provable results for Galois groups... however

                      CAVEAT EMPTOR!

 John McKay



On Mon, 24 Apr 2006, marta asaeda wrote:

> Hello
>
> I am wondering if galois group computation using gap is useable as a
> mathematical result. Is there any ambiguity ? Or, if I give a polynomial,
> and it computes its galois group, is it mathematically certain that the
> polynomial I gave was irreducible ? How does gap know that it is irreducible
> ?
>
> marta
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