Term Rewrite Rules for Finite Fields

Let F1, . . . ,Fk be finite fields with distinct characteristics. We give a finite set of equations which axiomatize the equational theory of F1, . . . ,Fk and then use these axioms to find a finite AC-term rewrite system which is complete for this theory. In particular this gives finite complete AC-term rewrite systems for many instances of x ≈ x rings. However there are m for which no such term rewrite system exists — the smallest is m = 22. A popular test problem for automated theorem proving has been the derivation of xy ≈ yx from ring axioms plus x ≈ x (m > 1). At present this has been achieved for m = 2, 3, 4, 6 using versions of the Knuth-Bendix algorithm. Zhang [6] has given a polynomial gcd algorithm which can be used to prove commutativity for about three-fourths of the even exponents m (by reducing such cases to x ≈ x or x ≈ x). A related problem is to find a complete set of AC-term rewrite rules for x ≈ x rings. This has been solved for m = 2 by Hsiang [2] and for m = 3 by Stickel [5]. As a corollary to our theorem we find such rewrite rules for many m ≥ 2 (about 94% of the m below 100,000). Our result also covers Nipkow’s [4] AC-term rewrite rules for rings satisfying x ≈ x and p ≈ 0, where p is a prime. 1 From Equations to Rewrite Rules. Throughout this section let F1, . . . ,Fk be a fixed list of finite fields with distinct characteristics p1, . . . , pk. The set of equations true of the fields Fi Research supported by a grant from NSERC Research supported by a grant from NSERC