Re: Computing Galois groups

[Daniel Herring]

Hi Fabio,

I don't have the time or energy to do a proper review for you.

That said, trying to explain an algorithm can be an effective way of 
finding the flaws in it.  Start with the code and derive the purpose. 
This often reveals an inconsistency with the original high-level 
algorithm.

Unfortunately, other bugs come from an incorrect understanding of the tool 
you are trying to use, in this case Axiom.  These can require an 
experienced developer to find, as you see what you expect, and they see 
what the computer actually does.  Sometimes you can reduce this issue by 
replacing binary operators with named functions (less parsing mistakes). 
Other times you can do "the same thing" with a different choice of 
functions or operators, and compare results.

Do you have a simple example of when the algorithm does not work? 
Tracing its progress step by step can help spot the bug.  This approach 
doesn't prove the overall algorithm, but it does eliminate one defect.

Best wishes!

-- Daniel


On Sat, 7 May 2022, Fabio Stumbo wrote:

> Hi,
>
> I am trying to compute some Galois groups with Axiom.
>
> A way is to go after the example given in section 8.13 of the manual which 
> follows closely and implements the original definition by Galois of the 
> Galois group.
>
> I am trying a different way: I would like to exploit a theorem by Kronecker 
> which provides an easy algorithm to compute the Galois group of a given 
> polynomial.
>
> I wrote the function which implements the algorithm and it works... 
> sometimes. :(
>
> Which means that I have some problems.
>
> My question is: is there anybody that is willing to help me to fix it if I 
> post the code?
>
> The code is not very long (about 40-50 lines) but it needs to be explained, 
> so I am asking before to make the effort.
>
> TIA
>
> Fabio