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Hi,

Python is object oriented (see Python doc about classes) and what you call functions are actually methods, i.e. they belong to their class. The point is that you can have different implementation of the objects and keep the same interface. In this particular case, given any function field you will use

sage: K.base_field()


to get the base field.

For the existence of the two classes, it is explicitly said in the documentation:

• FunctionField: The abstract base class for all function fields.
• RationalFunctionField: A rational function field K(t) in one variable, over an arbitrary base field.

Vincent