On Mon, Mar 26, 2012 at 4:41 AM, David Loeffler <[email protected]> wrote:
> Dear Ben,
>
> I'm afraid that Sage's number field functionality uses PARI heavily, and the
> PARI guys have made a policy decision not to try and support number fields
> defined by polynomials that aren't integral or aren't monic. This is one of
> the oldest tickets in the Sage bug tracker
> (http://trac.sagemath.org/sage_trac/ticket/252) -- it's been a recognised
> shortcoming of Sage for over 5 years.
>
> There is a function somewhere in sage/rings/qqbar.pyx called
> "clear_denominators" which can be used to find a monic integral polynomial
> which defines the same field as the original polynomial.
And another here:
sage: R.<z>=PolynomialRing(QQ)
sage: sage.schemes.elliptic_curves.heegner.make_monic(z^3 + 1/3)
(z^3 + 9, 3)
I don't think fixing number fields to work in general with non-monic
defining polys is really "that" hard via some sort of wrapper. It's
not like "rewrite abelian groups" where many people have tried and had
trouble. I don't think anybody has seriously tried.
William
>
> David
>
>
> On Monday, 26 March 2012 01:50:38 UTC+1, Ben wrote:
>>
>> I'm trying to work with some roots of a rational polynomial, but I can't
>> seem to get it to work. One example is
>>
>> z^3 + 1/3
>>
>> I seem to be able to get the splitting field from:
>>
>> R.<z>=PolynomialRing(QQ)
>> K.<a>=NumberField([z^3 + 1/3])
>> R.<z>=PolynomialRing(K)
>> print (z^3 + 1/3).factor()
>> L.<b>=NumberField([z^2 + a*z + a^2])
>>
>> But, I can't seem to do anything with it
>>
>> b^3 + 1/3
>>
>> returns
>>
>> Traceback (most recent call last):
>> File "<stdin>", line 1, in <module>
>> File "_sage_input_52.py", line 10, in <module>
>> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8
>> -*-\\n" +
>> _support_.preparse_worksheet_cell(base64.b64decode("Ui48ej49UG9seW5vbWlhbFJpbmcoUVEpCksuPGE+PU51bWJlckZpZWxkKFt6XjMgKyAxLzNdKQpSLjx6Pj1Qb2x5bm9taWFsUmluZyhLKQpwcmludCAoel4zICsgMS8zKS5mYWN0b3IoKQpMLjxiPj1OdW1iZXJGaWVsZChbel4yICsgYSp6ICsgYV4yXSkKYl4zKzEvMw=="),globals())+"\\n");
>> execfile(os.path.abspath("___code___.py"))
>> File "", line 1, in <module>
>>
>> File "/tmp/tmpTZzoeR/___code___.py", line 8, in <module>
>> exec compile(u'b**_sage_const_3 +_sage_const_1 /_sage_const_3
>> File "", line 1, in <module>
>>
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/displayhook.py",
>> line 174, in displayhook
>> print_obj(sys.stdout, obj)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/displayhook.py",
>> line 142, in print_obj
>> print >>out_stream, `obj`
>> File "sage_object.pyx", line 154, in
>> sage.structure.sage_object.SageObject.__repr__
>> (sage/structure/sage_object.c:1492)
>> File "number_field_element.pyx", line 3760, in
>> sage.rings.number_field.number_field_element.NumberFieldElement_relative._repr_
>> (sage/rings/number_field/number_field_element.cpp:22839)
>> File "number_field_element.pyx", line 3727, in
>> sage.rings.number_field.number_field_element.NumberFieldElement_relative.list
>> (sage/rings/number_field/number_field_element.cpp:22612)
>> File "number_field_element.pyx", line 2521, in
>> sage.rings.number_field.number_field_element.NumberFieldElement.vector
>> (sage/rings/number_field/number_field_element.cpp:17386)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field_rel.py",
>> line 1210, in relative_vector_space
>> from_V = maps.MapRelativeVectorSpaceToRelativeNumberField(V, self)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/maps.py",
>> line 278, in __init__
>> self.__rnf = K.pari_rnf()
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/cachefunc.py",
>> line 555, in __call__
>> w = self._cachedmethod._instance_call(self._instance, *args, **kwds)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/cachefunc.py",
>> line 778, in _instance_call
>> return self._cachedfunc.f(inst, *args, **kwds)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field_rel.py",
>> line 1398, in pari_rnf
>> return self._pari_base_nf().rnfinit(self.pari_relative_polynomial())
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/cachefunc.py",
>> line 555, in __call__
>> w = self._cachedmethod._instance_call(self._instance, *args, **kwds)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/misc/cachefunc.py",
>> line 778, in _instance_call
>> return self._cachedfunc.f(inst, *args, **kwds)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field_rel.py",
>> line 1116, in _pari_base_nf
>> return abs_base.pari_nf()
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 2677, in pari_nf
>> raise TypeError, "Unable to coerce number field defined by
>> non-integral polynomial to PARI."
>> TypeError: Unable to coerce number field defined by non-integral
>> polynomial to PARI.
>>
>>
>>
>> Trying instead to work with a galois_closure I can't even define the field
>>
>> R.<z>=PolynomialRing(QQ)
>> K.<a>=NumberField([z^3 + 1/3])
>> L.<b>=K.galois_closure()
>>
>> I get
>>
>> Traceback (most recent call last):
>> File "<stdin>", line 1, in <module>
>> File "_sage_input_53.py", line 10, in <module>
>> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8
>> -*-\\n" +
>> _support_.preparse_worksheet_cell(base64.b64decode("Ui48ej49UG9seW5vbWlhbFJpbmcoUVEpCksuPGE+PU51bWJlckZpZWxkKFt6XjMgKyAxLzNdKQpMLjxiPj1LLmdhbG9pc19jbG9zdXJlKCkKYl4zKzEvMw=="),globals())+"\\n");
>> execfile(os.path.abspath("___code___.py"))
>> File "", line 1, in <module>
>>
>> File "/tmp/tmpbWZl2Z/___code___.py", line 5, in <module>
>> L = K.galois_closure(names=('b',)); (b,) = L._first_ngens(1)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 6161, in galois_closure
>> L, self_into_L = self._galois_closure_and_embedding(names)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 6094, in _galois_closure_and_embedding
>> newK, K_into_newK, _, _ = K.composite_fields(self, names=names,
>> both_maps=True, preserve_embedding=False)[-1]
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 3469, in composite_fields
>> self_to_F = self.hom([F(a_in_F)])
>> File "parent.pyx", line 988, in sage.structure.parent.Parent.__call__
>> (sage/structure/parent.c:7355)
>> File "coerce_maps.pyx", line 82, in
>> sage.structure.coerce_maps.DefaultConvertMap_unique._call_
>> (sage/structure/coerce_maps.c:3311)
>> File "coerce_maps.pyx", line 77, in
>> sage.structure.coerce_maps.DefaultConvertMap_unique._call_
>> (sage/structure/coerce_maps.c:3214)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 1226, in _element_constructor_
>> return self._coerce_non_number_field_element_in(x)
>> File
>> "/home/ben/sage-4.8/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py",
>> line 5376, in _coerce_non_number_field_element_in
>> return self._element_class(self, x)
>> File "number_field_element.pyx", line 340, in
>> sage.rings.number_field.number_field_element.NumberFieldElement.__init__
>> (sage/rings/number_field/number_field_element.cpp:5545)
>> TypeError: Coercion of PARI polmod with modulus 243*x^6 - 1620*x^3 + 9261
>> into number field with defining polynomial 9*x^6 - 60*x^3 + 343 failed
>>
>>
>>
>> I'd like to be able to work algebraically with the roots of this
>> polynomial. Is there a way to get this to work?
>>
>> Thanks,
>> Ben
>
> --
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
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