#4276: [with patch, needs review] move number fields to new coercion, implement
embeddings
----------------------+-----------------------------------------------------
Reporter: robertwb | Owner: robertwb
Type: defect | Status: new
Priority: major | Milestone: sage-3.2.2
Component: coercion | Resolution:
Keywords: |
----------------------+-----------------------------------------------------
Comment (by ncalexan):
I am trying to referee this, but it's going to take some time. I'm going
to put it into my tree and try to move some of my code that does this by
hand to the embedding architecture. Email me in a month.
I have some questions and problems:
I really need to be able to specify embeddings after creation, because
*tons* of functions create number fields (composite_fields, for example,
or subfields) and I really don't want to specify all possible embeddings
of the results...
First, why squarefree_part bounded? This could be a separate patch.
Second, I can specify non-embeddings that seem to be ignored:
{{{
sage: K.<a> = NumberField(x^3+2, embedding=CC(-10).nth_root(3))
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^3 + 2
To: Complex Lazy Field
Defn: a -> 0.6299605249474365? + 1.091123635971722?*I
sage: K.gen_embedding()^3
-2.00000000000000? + 0.?e-14*I
}}}
Finally, I am having lots of trouble creating non-default embeddings:
{{{
sage: K2.<a> = NumberField(x^2 + 2, 'a', embedding=RR(-sqrt(2)))
---------------------------------------------------------------------------
IndexError Traceback (most recent call
last)
/Users/ncalexan/.sage/temp/mero.local/25160/_Users_ncalexan__sage_init_sage_0.py
in <module>()
----> 1
2
3
4
5
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in
NumberField(polynomial, name, check, names, cache, embedding)
369
370 if polynomial.degree() == 2:
--> 371 K = NumberField_quadratic(polynomial, name, check,
embedding)
372 else:
373 K = NumberField_absolute(polynomial, name, None, check,
embedding)
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in __init__(self,
polynomial, name, check, embedding)
6470 Number Field in a with defining polynomial x^2 - 4
6471 """
-> 6472 NumberField_absolute.__init__(self, polynomial, name=name,
check=check, embedding=embedding)
6473 self._element_class =
number_field_element_quadratic.NumberFieldElement_quadratic
6474 c, b, a = [rational.Rational(t) for t in
self.defining_polynomial().list()]
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in __init__(self,
polynomial, name, latex_name, check, embedding)
3649 NumberField_generic.__init__(self, polynomial, name,
latex_name, check, embedding)
3650 self._element_class =
number_field_element.NumberFieldElement_absolute
-> 3651 self._zero_element = self(0)
3652 self._one_element = self(1)
3653
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in sage.structure.parent.Parent.__call__
(sage/structure/parent.c:3665)()
277
278
--> 279
280
281
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.convert_map_from
(sage/structure/parent.c:8640)()
1000
1001
-> 1002
1003
1004
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.discover_convert_map_from
(sage/structure/parent.c:8771)()
1007
1008
-> 1009
1010
1011
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.coerce_map_from
(sage/structure/parent.c:7653)()
876
877
--> 878
879
880
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.discover_coerce_map_from
(sage/structure/parent.c:7956)()
921
922
--> 923
924
925
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent_old.so in
sage.structure.parent_old.Parent._coerce_map_from_
(sage/structure/parent_old.c:6291)()
576
577
--> 578
579
580
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in
_coerce_map_from_(self, K)
6497 return
number_field_element_quadratic.Q_to_quadratic_field_element(self)
6498 else:
-> 6499 return NumberField_absolute._coerce_map_from_(self, K)
6500
6501
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in
_coerce_map_from_(self, R)
1525 """
1526 if R in [int, long, ZZ, QQ, self.base()]:
-> 1527 return self._generic_convert_map(R)
1528 from sage.rings.number_field.order import
is_NumberFieldOrder
1529 if is_NumberFieldOrder(R) and
R.number_field().has_coerce_map_from(self):
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent_old.so in
sage.structure.parent_old.Parent._generic_convert_map
(sage/structure/parent_old.c:6841)()
594
595
--> 596
597
598
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent_old.so in
sage.structure.parent_old.Parent._generic_convert_map
(sage/structure/parent_old.c:6806)()
602
603
--> 604
605
606
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent._generic_convert_map
(sage/structure/parent.c:6875)()
747
748
--> 749
750
751
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/coerce_maps.so in
sage.structure.coerce_maps.DefaultConvertMap.__init__
(sage/structure/coerce_maps.c:1931)()
22
23
---> 24
25
26
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/categories/map.so in sage.categories.map.Map.__init__
(sage/categories/map.c:1768)()
39
40
---> 41
42
43
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/categories/homset.pyc in Hom(X, Y, cat)
151
152 ##_cache[key] = weakref.ref(H)
--> 153 _cache[(X, Y, cat)] = weakref.ref(H)
154
155 return H
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/number_field/number_field.pyc in __cmp__(self, other)
1639 else:
1640 return cmp(self.coerce_embedding()(self.gen()),
-> 1641 other.coerce_embedding()(other.gen()))
1642
1643 def _ideal_class_(self):
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyBinop.__richcmp__ (sage/rings/real_lazy.c:7858)()
868
869
--> 870
871
872
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/element.so in
sage.structure.element.Element._richcmp (sage/structure/element.c:5169)()
530
531
--> 532
533
534
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/element.so in
sage.structure.element.Element._richcmp_c_impl
(sage/structure/element.c:5400)()
577
578
--> 579
580
581
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyFieldElement._cmp_c_impl
(sage/rings/real_lazy.c:5445)()
508
509
--> 510
511
512
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyFieldElement.approx
(sage/rings/real_lazy.c:5880)()
567
568
--> 569
570
571
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in sage.rings.real_lazy.LazyBinop.eval
(sage/rings/real_lazy.c:7427)()
825
826
--> 827
828
829
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in sage.rings.real_lazy.LazyBinop.eval
(sage/rings/real_lazy.c:7439)()
826
827
--> 828
829
830
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyAlgebraic.eval (sage/rings/real_lazy.c:10310)()
1278
1279
-> 1280
1281
1282
IndexError: list index out of range
sage:
}}}
And even worse, non-embeddings. check the sign of sqrt(2) below)
{{{
sage: K1.<a> = NumberField(x^2 + 2, 'a', embedding=RR(sqrt(2)))
sage: K1.gen_embedding()
---------------------------------------------------------------------------
IndexError Traceback (most recent call
last)
/Users/ncalexan/.sage/temp/mero.local/25160/_Users_ncalexan__sage_init_sage_0.py
in <module>()
----> 1
2
3
4
5
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/IPython/Prompts.pyc in __call__(self, arg)
549
550 # and now call a possibly user-defined print mechanism
--> 551 manipulated_val = self.display(arg)
552
553 # user display hooks can change the variable to be
stored in
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/IPython/Prompts.pyc in _display(self, arg)
575 return IPython.generics.result_display(arg)
576 except TryNext:
--> 577 return self.shell.hooks.result_display(arg)
578
579 # Assign the default display method:
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/IPython/hooks.pyc in __call__(self, *args, **kw)
133 #print "prio",prio,"cmd",cmd #dbg
134 try:
--> 135 ret = cmd(*args, **kw)
136 return ret
137 except ipapi.TryNext, exc:
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/IPython/hooks.pyc in result_display(self, arg)
163
164 if self.rc.pprint:
--> 165 out = pformat(arg)
166 if '\n' in out:
167 # So that multi-line strings line up with the left
column of
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python/pprint.pyc in
pformat(self, object)
109 def pformat(self, object):
110 sio = _StringIO()
--> 111 self._format(object, sio, 0, 0, {}, 0)
112 return sio.getvalue()
113
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python/pprint.pyc in
_format(self, object, stream, indent, allowance, context, level)
127 self._readable = False
128 return
--> 129 rep = self._repr(object, context, level - 1)
130 typ = _type(object)
131 sepLines = _len(rep) > (self._width - 1 - indent -
allowance)
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python/pprint.pyc in
_repr(self, object, context, level)
193 def _repr(self, object, context, level):
194 repr, readable, recursive = self.format(object,
context.copy(),
--> 195 self._depth,
level)
196 if not readable:
197 self._readable = False
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python/pprint.pyc in
format(self, object, context, maxlevels, level)
205 and whether the object represents a recursive construct.
206 """
--> 207 return _safe_repr(object, context, maxlevels, level)
208
209
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python/pprint.pyc in
_safe_repr(object, context, maxlevels, level)
290 return format % _commajoin(components), readable,
recursive
291
--> 292 rep = repr(object)
293 return rep, (rep and not rep.startswith('<')), False
294
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/structure/sage_object.so in
sage.structure.sage_object.SageObject.__repr__
(sage/structure/sage_object.c:1090)()
90
91
---> 92
93
94
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyFieldElement._repr_
(sage/rings/real_lazy.c:5833)()
555
556
--> 557
558
559
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyFieldElement.approx
(sage/rings/real_lazy.c:5880)()
567
568
--> 569
570
571
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in sage.rings.real_lazy.LazyBinop.eval
(sage/rings/real_lazy.c:7427)()
825
826
--> 827
828
829
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in sage.rings.real_lazy.LazyBinop.eval
(sage/rings/real_lazy.c:7439)()
826
827
--> 828
829
830
/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-
packages/sage/rings/real_lazy.so in
sage.rings.real_lazy.LazyAlgebraic.eval (sage/rings/real_lazy.c:10310)()
1278
1279
-> 1280
1281
1282
IndexError: list index out of range
sage:
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4276#comment:10>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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