#13379: Add a splitting field function for polynomials over a finite field
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Reporter: abrochard | Owner: davidloeffler
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.9
Component: number fields | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by mmasdeu):
Yes, I agree that for polynomials defined over F,,p,, the function should
be simplified. As for F,,q,, one definitely needs to fix the code, since
that calling of Hom won't work...
Replying to [comment:7 robharron]:
> But have you looked at the code? It doesn't do anything special with
regards to the embedding. (Do you see something I don't?) In the end it
simply says F.Hom(K)[0] (for instance, I think the code only runs when the
polynomial is defined over a prime field, in which case the embedding
returned is simply the canonical one sending 1 to 1). I just quickly wrote
a function that does what I suggested and it works just fine and is quite
quick.
>
> Replying to [comment:6 mmasdeu]:
> > I think that the whole point is to return an embedding of F,,q,, into
its splitting field. For that, one has to be careful when constructing the
extensions: either choosing the given polynomial (or a factor of it) to
define the extension, or using some root finding algorithm.
> >
> > Also, the patch fails in the last couple of Sage releases...
> >
> > Replying to [comment:5 robharron]:
> > > For p a prime and q = p^n^, the field F,,q,, is Galois over F,,p,,,
i.e. it is the splitting field of any irreducible polynomial of degree n
over F,,p,,. Moreover, the field with p^n^ elements contains the one with
p^m^ elements if and only if m divides n. Thus, starting from a polynomial
over F,,p,,, the splitting field is simply given by taking the lcm of the
degrees of the factors. This would be quicker than what you are doing. If
you start with a polynomial over F,,q,,, a simple modification of what
I've said would work too.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13379#comment:8>
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