I agree that this would be a useful funtion to have. I would call it
splitting_field() with a description similar to that of root_field()
-- whose docstring does not say that self should be irreducible,
though in fact it must.
In the meantim you should be able to work with what is available as follows:
sage: QQx.<x>=QQ[]
sage: f=(x^2-2)*(x^2-3)
sage: F=NumberField([p for p,n in f.factor()],'a')
sage: F2=F.absolute_field('b')
sage: F2.structure()
(Isomorphism from Number Field in b with defining polynomial x^4 -
10*x^2 + 1 to Number Field in a0 with defining polynomial x^2 - 3 over
its base field,
Isomorphism from Number Field in a0 with defining polynomial x^2 - 3
over its base field to Number Field in b with defining polynomial x^4
- 10*x^2 + 1)
Here F is first defined as a relative extension, with generators a0,a1
satisfying the equations:
sage: a0,a1=F.gens()
sage: a0^2, a1^2
(3, 2)
then F2 is the associated absolute field, with F2.structure() giving
maps from each of these into the other.
sage: F2toF, FtoF2=F2.structure()
sage: FtoF2(a0)
-1/2*b^3 + 11/2*b
sage: FtoF2(a0).minpoly()
x^2 - 3
sage: FtoF2(a1)
-1/2*b^3 + 9/2*b
sage: FtoF2(a1).minpoly()
x^2 - 2
John
On 16/02/2008, Jason Grout <[EMAIL PROTECTED]> wrote:
>
> John Cremona wrote:
> > Are you sure you mean to give NumberField() two polynomials, one of
> > which (x) defines the trivial extension? You are only giving one name
> > so I rpresume what you mean (to define a quadratic field) is
> >
> >
> > sage: NumberField([x^2-3],'a')
> > Number Field in a with defining polynomial x^2 - 3
> > sage: F=NumberField([x^2-3],'a')
> > sage: F.gens()
> > (a,)
>
> Yes, I'd like to get the quadratic extension. I know I can do this
> without the x; my question was about why it didn't work the way I was
> trying (and why it was complaining that x was not irreducible), but I
> didn't know if I was naively doing something wrong either.
>
> What I'm trying to do is get a number field that has all the roots of a
> (not necessarily irreducible) polynomial. I was trying to automatically
> construct a field that contained the eigenvalues of a matrix. But when
> calling NumberField on the list of irreducible factors of the
> characteristic polynomial, things were blowing or not making sense to me
> if there was a linear factor somewhere.
>
> Is there an easier way to get an extension that contains all the roots
> of a given polynomial (which may not be irreducible)? I tried the
> following, but it didn't work; should it?
>
> sage: a=QQ['x']; x=a.gen()
> sage: b=(x^2-2)*(x^2-3)
> sage: b.root_field('y')
> ---------------------------------------------------------------------------
> <type 'exceptions.ValueError'> Traceback (most recent call last)
>
> /home/grout/<ipython console> in <module>()
>
> /home/grout/polynomial_element.pyx in
> sage.rings.polynomial.polynomial_element.Polynomial.root_field()
>
> /home/grout/sage/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py
> in NumberField(polynomial, name, check, names, cache)
> 274 K = NumberField_quadratic(polynomial, name, check)
> 275 else:
> --> 276 K = NumberField_absolute(polynomial, name, None, check)
> 277
> 278 if cache:
>
> /home/grout/sage/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py
> in __init__(self, polynomial, name, latex_name, check)
> 2466
> 2467 def __init__(self, polynomial, name, latex_name=None,
> check=True):
> -> 2468 NumberField_generic.__init__(self, polynomial, name,
> latex_name, check)
> 2469 self._element_class =
> number_field_element.NumberFieldElement_absolute
> 2470
>
> /home/grout/sage/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py
> in __init__(self, polynomial, name, latex_name, check)
> 621 if check:
> 622 if not polynomial.is_irreducible():
> --> 623 raise ValueError, "defining polynomial (%s) must
> be irreducible"%polynomial
> 624 if not polynomial.parent().base_ring() == QQ:
> 625 raise TypeError, "polynomial must be defined
> over rational field"
>
> <type 'exceptions.ValueError'>: defining polynomial (x^4 - 5*x^2 + 6)
> must be irreducible
>
>
>
> >
> > On 16/02/2008, Jason Grout <[EMAIL PROTECTED]> wrote:
> >> Is the following output for b.gens() correct?
> >>
> >> sage: NumberField([x,x^2-3],'a')
> >> Number Field in a0 with defining polynomial x over its base field
> >> sage: b=NumberField([x,x^2-3],'a')
> >> sage: b.gens()
> >> (0, 0)
> >>
> >> To contrast:
> >>
> >> sage: c=NumberField([x^2-3, x^2-2],'a')
> >> sage: c.gens()
> >> (a0, a1)
> >>
> >>
> >> Also, this blows up:
> >>
> >> sage: c=NumberField([x^2-3, x],'a')
> >
> > The problem here is that x is triggering a an error in the
> > irreducibility test, which is a little bizarre since of course x is
> > irreducible.
> >
> > So the real issue is: why is x allowed to determine an absolute number
> > field (base Q) but not a relative one? My guess is that this is a
> > side-effect of the differing code being used to test irreducibility in
> > the two cases,
> >
> > Personally, I think that trivial extensions should be allowed and
> > treated just as non-trivial ones. I have recently had to define
> > extensions of the ring ZZ, and find this awkward:
> >
> > sage: R=ZZ.extension(x^2+5,'a')
> > sage: R.gens()
> > [1, a]
> > sage: S=ZZ.extension(x+5,'b')
> > sage: S.gens()
> > [1]
> >
> > In the latter case I need S to remember the polynomial used to
> > generaite it and would expect its gens() to include (in this case) -5.
> >
> > On the same topic, R and S above have no defining_polynomial() method.
> > I'll try to fix that if it looks easy.
>
> Thanks!
>
> Jason
>
>
> >
>
--
John Cremona
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-devel@googlegroups.com
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at http://groups.google.com/group/sage-devel
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
