Simultaneous embedding of CyclotomicField(n) and sqrt(n)

[GaryMak]

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 ...

[John Cremona]

Add this line after your code:

sage: CC.register_coercion(L.embeddings(CC)[0])

(note that L.embeddings(CC) is a list of all 4 embeddings. You may
prefer one of the others.) Now we can do

sage: CC(s)
-1.73205080756888

which is possibly not the sqrt(3) you had in mind. I don't know if it
is possible to "forget" that coercion without redefining L:

sage: L.<s> = Z.extension(X^2-n)
sage: CC.register_coercion(L.embeddings(CC)[2])
sage: CC(s)
1.73205080756888 - 1.11022302462516e-16*I

John

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[Gary McConnell]

Aha - not a command I had managed to discover - thank you very much - that fixes it!! (interestingly, as you say in your aside, if it has even a "broken" idea already of an embedding (eg my failed original one) then even that command of yours will cause an error screed - one seemingly needs to restart the program entirely or else, as you say, define L anew ... )

best regards

Gary

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