A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. Below we offer another proof of this result of Tarski. The main point of our proof is accomplished upon showing how to decide whether a given polynomial f(x, y) in two variables, defined over the field R of rational numbers, has a zero in a real-closed field K containing R.1 This is done in ?2, but for… Expand