There seems to be a bug in the implementation of the coercion from
universal cyclotomic field. The rational part disappears when converting.
For example:
F.<c>=CyclotomicField(4)
F(-1/2 + 1/2*E(4))
1/2*c^2
Reading the code:
from sage.rings.number_field.number_field import CyclotomicField
k = self._obj.Conductor().sage()
Rcan = CyclotomicField(k)
if R is None:
R = Rcan
obj = self._obj
if obj.IsRat():
return R(obj.sage())
zeta = Rcan.gen()
coeffs = obj.CoeffsCyc(k).sage()
return R(sum(coeffs[a] * zeta**a for a in range(1,k)))
I believe the problem is in the range(1,k) which should be range(0,k)
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