#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-5.8
Component: number fields | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by 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:6>
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