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Here are two ways to solve your first problem.

• use the Python version of the complex imaginary unit, 1j.

sage: R.<x,y> = CC['x,y'] sage: f = x + 1jy sage: f x + 1.00000000000000I*y sage: f.parent() Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision

• redefine I to be the imaginary unit in CC.

sage: I = CC.gen() sage: I = CC.gen() sage: f = x + I * y sage: f x + 1.00000000000000Iy sage: f.parent() Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision

Regarding your second problem, the following fails because R is not a principal ideal domain.

sage: span([f], R)


Here are two ways to solve your first problem.

• use the Python version of the complex imaginary unit, 1j.

sage: R.<x,y> = CC['x,y']
sage:  f = x + 1jy
1j*y
sage: f
x + 1.00000000000000I*y
1.00000000000000*I*y
sage: f.parent()
Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precisionprecision

• redefine I to be the imaginary unit in CC.

sage: I = CC.gen()
sage: I = CC.gen()
sage: f = x + I * y
sage: f
x + 1.00000000000000Iy
1.00000000000000*I*y
sage: f.parent()
Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precisionprecision


Regarding your second problem, the following fails because R is not a principal ideal domain.

sage: span([f], R)