+ # A manual dictionary-cache for the complex_extension() method,
+ # since apparently @classmethods can't also be @cached_methods.
+ _complex_extension = {}
+
+ @classmethod
+ def complex_extension(cls,field):
+ r"""
+ The complex field that we embed/unembed, as an extension
+ of the given ``field``.
+ """
+ if field in cls._complex_extension:
+ return cls._complex_extension[field]
+
+ # Sage doesn't know how to adjoin the complex "i" (the root of
+ # x^2 + 1) to a field in a general way. Here, we just enumerate
+ # all of the cases that I have cared to support so far.
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ elif not field.is_exact():
+ # RDF or RR
+ F = field.complex_field()
+ else:
+ # Works for QQ and... maybe some other fields.
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+
+ cls._complex_extension[field] = F
+ return F
+