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Euclid's process for finding the G.C.D. of two integers a, 6 leads to the G.C.D. of F{a), F{b). We call F (a) a prime if its only divisors are F(zk a) and F(db 1). When the preceding equation holds, we say that F (a) and F(r) are congruent modulo F(b). It is now a simple matter to enunciate the analogues, for the numbers F (a) and the operations © and 0 , of the theorems in the theory of numbers. If p is a prime not dividing a, then F (a) o F (a) • • • o F (a), to p — 1 factors, is congruent to ^(1) modulo F(p). Again, F(l) 0 F(2) • • • o F(p 1) is congruent to F(— 1). These analogues to Fermat's and Wilson's theorems follow at once

An extension of the theory of numbers by means of correspondences between fields