In Chapter I, we deal exclusively with certain elementary arithwetical properties of Z which carry over directly as meaningful properties in GF [q,Xi]. In Chapter II, where we assume k = 1, we discuss problems involving irreducibility of polynomials, which, at first sight, have no analogue in Z. However, these problems are essentially ones concerned with the decomposition of primes of an algebraic function field in a finite extension field and so with the basic problem of algebraic number theory. Although we give a partial treatment of our problem without explicitly mentioning this theory, we also develop it along algebraic number theoretical lines sufficiently to show the connection.