Hi everyone
I am doing some calcs involving n-th roots of unity and the square root of
n, all viewed ultimately as complex numbers (where n is any positive
integer). Everything is of course fine if sqrt(n) is in QQ(zeta_n), and
everything is also fine for other n except that if I wish to step out of
the number field situation and evaluate things as complexes, I begin to get
embedding errors which I do not understand. I have tried at least a dozen
completely different approaches based on the manual, but in vain. In short,
here is an example:
n = 3
Z.<z> = CyclotomicField(n)
LX.<X> = Z[]
L.<s> = Z.extension(X^2-n)
Now, if I check that everything is in order inside L, it all seems fine at
this stage - for example
s in L
z in L
both return 'True' and moreover both behave as per their defining
equations. The problem arises as soon as I try to manipulate anything
involving 's' as a complex number, for example even the trivial
CC(s)
returns
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_59.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n"
+
_support_.preparse_worksheet_cell(base64.b64decode("Q0Mocyk="),globals())+"\\n");
execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpfVbXx5/___code___.py", line 2, in <module>
exec compile(u'CC(s)
File "", line 1, in <module>
File
"/home/sage/sage-5.7/local/lib/python2.7/site-packages/sage/rings/complex_field.py",
line 277, in __call__
return Parent.__call__(self, x)
File "parent.pyx", line 834, in sage.structure.parent.Parent.__call__
(sage/structure/parent.c:7415)
File "coerce_maps.pyx", line 82, in
sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(sage/structure/coerce_maps.c:3583)
File "coerce_maps.pyx", line 77, in
sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(sage/structure/coerce_maps.c:3485)
File
"/home/sage/sage-5.7/local/lib/python2.7/site-packages/sage/rings/complex_field.py",
line 308, in _element_constructor_
return complex_number.ComplexNumber(self, x)
File "complex_number.pyx", line 162, in
sage.rings.complex_number.ComplexNumber.__init__
(sage/rings/complex_number.c:3647)
File "parent.pyx", line 834, in sage.structure.parent.Parent.__call__
(sage/structure/parent.c:7415)
File "coerce_maps.pyx", line 156, in
sage.structure.coerce_maps.NamedConvertMap._call_
(sage/structure/coerce_maps.c:4593)
File "number_field_element.pyx", line 1333, in
sage.rings.number_field.number_field_element.NumberFieldElement._mpfr_
(sage/rings/number_field/number_field_element.cpp:11441)
File "parent.pyx", line 834, in sage.structure.parent.Parent.__call__
(sage/structure/parent.c:7415)
File "coerce_maps.pyx", line 82, in
sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(sage/structure/coerce_maps.c:3583)
File "coerce_maps.pyx", line 77, in
sage.structure.coerce_maps.DefaultConvertMap_unique._call_
(sage/structure/coerce_maps.c:3485)
File
"/home/sage/sage-5.7/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py",
line 8058, in _element_constructor_
return NumberField_absolute._element_constructor_(self, x)
File
"/home/sage/sage-5.7/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py",
line 1230, in _element_constructor_
return self._coerce_from_other_number_field(x)
File
"/home/sage/sage-5.7/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py",
line 5671, in _coerce_from_other_number_field
raise ValueError, "Cannot convert %s to %s (regardless of embeddings)"%(x,K)
ValueError: Cannot convert s to Cyclotomic Field of order 3 and degree 2
(regardless of embeddings)
Of course I would understand this if I were trying to coerce s naively into
Z, but I cannot understand why this is happening in this situation (since
SAGE seems quite happy with s being a member of L, ergo I would have
thought that it would have some appropriate complex embedding 'in mind'
...). I have tried to use embeddings and extensions in various guises, but
to no avail. Also I would like to use the machinery SAGE has for handling
cyclotomic numbers, so I would prefer not to have to define this by first
looking up the cyclotomic polynomial and defining the number field using
two explicit equations etc.
Many thanks for any suggestions ...
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