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Square root in FunctionField


I'm working with $q$-functions and I would like to define a free algebra over the algebraic field with an extra parameter $q$ for which I can also take roots i.e. I would like to do something like this

sage: F.<q> = FunctionAlgebra(AA)
sage: A.<a,b,c,d> = FreeAlgebra(F, 4)
sage: (1+q)**(1/2)*a*b

The function FunctionAlgebra is not the correct function, since it only includes (integer) powers of $q$. Is it possible to extend this FunctionAlgebra to something where sums of powers of $q$ also have roots? Or is there an other function for which I can do this?

Best, Noud