We analyse the relations between completion procedures for polynomials and terms and thereby show how Buchberger's algorithm for multivariate polynomials over finite fields and over the rationals can be simulated using term completion modulo AC. To specify the rational numbers an infinite term rewriting system is needed. However, for the simulation of each particular ideal completion a finite approximation of the infinite rule set is sufficient. This approximation can be constructed during the

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... ompletion. The division operation needed in Buchberger's algorithm reduces to a narrowing procedure which becomes part of the critical pair computation process.