Abstract: In this paper, minimum realization of transfer function is discussed with algebraic means. Starting from the concept of whether or not the transfer function can be made common denomination, a new algebraic criterion of controllability and observability is introduced. State formulae of the ladder network and structure controllability and observability, algebraic means of any minimum realization by state feedback and output are also discussed. An example is given foF illustration. Compared with state space method, this means of realization is simple in structure and uses minimum active and passive elements, and the values of the elements in the ladder network can be determined arbitrarily. Various values of gain for the adder can be obtained by simple calculation.