def _element_constructor_(self, elt): r""" TESTS:: sage: UCF = UniversalCyclotomicField() sage: UCF(3) 3 sage: UCF(3/2) 3/2 sage: C = CyclotomicField(13) sage: UCF(C.gen()) E(13) sage: UCF(C.gen() - 3*C.gen()**2 + 5*C.gen()**5) E(13) - 3*E(13)^2 + 5*E(13)^5 sage: C = CyclotomicField(12) sage: zeta12 = C.gen() sage: a = UCF(zeta12 - 3* zeta12**2) sage: a -E(12)^7 + 3*E(12)^8 sage: C(_) == a True sage: UCF('[[0, 1], [0, 2]]') Traceback (most recent call last): ... TypeError: [ [ 0, 1 ], [ 0, 2 ] ] of type <type 'sage.libs.gap.element.GapElement_List'> not valid to initialize an element of the universal cyclotomic field Some conversions from symbolic functions are possible:: sage: UCF = UniversalCyclotomicField() sage: [UCF(sin(pi/k, hold=True)) for k in range(1,10)] [0, 1, -1/2*E(12)^7 + 1/2*E(12)^11, 1/2*E(8) - 1/2*E(8)^3, -1/2*E(20)^13 + 1/2*E(20)^17, 1/2, -1/2*E(28)^19 + 1/2*E(28)^23, 1/2*E(16)^3 - 1/2*E(16)^5, -1/2*E(36)^25 + 1/2*E(36)^29] sage: [UCF(cos(pi/k, hold=True)) for k in range(1,10)] [-1, 0, 1/2, 1/2*E(8) - 1/2*E(8)^3, -1/2*E(5)^2 - 1/2*E(5)^3, -1/2*E(12)^7 + 1/2*E(12)^11, -1/2*E(7)^3 - 1/2*E(7)^4, 1/2*E(16) - 1/2*E(16)^7, -1/2*E(9)^4 - 1/2*E(9)^5] .. TODO:: Implement conversion from QQbar (and as a consequence from the symbolic ring) """ elt = py_scalar_to_element(elt) if isinstance(elt, (Integer, Rational)): return self.element_class(self, libgap(elt)) elif isinstance( elt, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)): return self.element_class(self, elt) elif not elt: return self.zero() obj = None if isinstance(elt, gap.GapElement): obj = libgap(elt) elif isinstance(elt, gap3.GAP3Element): obj = libgap.eval(str(elt)) elif isinstance(elt, str): obj = libgap.eval(elt) if obj is not None: if not isinstance(obj, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)): raise TypeError( "{} of type {} not valid to initialize an element of the universal cyclotomic field" .format(obj, type(obj))) return self.element_class(self, obj) # late import to avoid slowing down the above conversions from sage.rings.number_field.number_field_element import NumberFieldElement from sage.rings.number_field.number_field import NumberField_cyclotomic, CyclotomicField P = parent(elt) if isinstance(elt, NumberFieldElement) and isinstance( P, NumberField_cyclotomic): n = P.gen().multiplicative_order() elt = CyclotomicField(n)(elt) return sum(c * self.gen(n, i) for i, c in enumerate(elt._coefficients())) if hasattr(elt, '_algebraic_'): return elt._algebraic_(self) raise TypeError( "{} of type {} not valid to initialize an element of the universal cyclotomic field" .format(elt, type(elt)))