Generators in the field of hyperelliptic functions

We consider the field of hyperelliptic functions defined for a family of hyperelliptic curves as rational functions in some special functions from Kleinian functions theory. We compare our definition with the classical one. We provide details and references for the result that the field of hyperelliptic functions for a family of hyperelliptic curves of genus $g$ is isomorphic to the field of rational functions with $3g$ generators. The main result of the present work is that there are no algebraic relations between these generators.