Recent interest in the mechanization of operations in polynomial algebra has led to the production of programs such as ALPAK, FORMAC, Formula ALGOL (with associated translator), GRAD Assistant and PM. (See the survey in Comm. ACM, Vol. 9, No. 8, August 1966; and Brown, Hyde and Tague, 1962-3, Williams, 1962, and Collins, 1966a.) These programs provide for the handling of polynomials in one or several variables and for operations on them such as addition, subtraction, multiplication, differentiation, substitution for variables, and calculation of residues. The aim of these programs has usually been simply to provide assistance in the (laborious and error-prone) task of calculating with algebraic expressions. More ambitious aims are possible, however, since the economical mechanization of certain systematic approaches to algebra, such as that provided by Tarski's decision method (Tarski, 1948; Seidenberg, 1954; Collins, 19666), could open up an entirely new field to the computer.