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Given an unstable finite-dimensional linear system, the output feedback problem is, first, to decide whether it is possible by memoryless linear feedback of the output to stabilize the system, and, second, to determine a stabilizing feedback law if such exists. This paper shows how this and a number of other linear system theory problems can be simply reformulated so as to allow application of known algorithms for solution of the existence question, with the construction problem being solved by some extension of these known algorithms. The first part of the output feedback problem is solvable with a finite number of rational operations, and the second with a finite number of polynomial factorizations. Other areas of application of the algorithm are described: Stability and positivity tests, low-order observer and controller design, and problems related to output feedback. Alternative computational procedures more or less divorced from the known algorithms are also proposed.