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.
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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