This edition has been updated to incorporate new approaches on

p-adic integers and modules and on the determinability of a module by its automorphism group.

Throughout this paper, [Z.sub.p], [Q.sub.p], and [C.sub.p] denote the ring of

p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of [Q.sub.p], respectively.

is the unit disk of [??], consisting of all

p-adic integers.

If p is an odd prime, then every integer n [equivalent to] 1 mod [p.sup.1/p(g)+1] is a gth power in the ring [Z.sub.p] of

p-adic integers. The same holds if p = 2 under the stronger hypothesis that n = 1 mod p[sup.[nu]p(g)+2].

These modules play important roles in various areas of algebra, primarily commutative algebra, including rings of

p-adic integers and certain power series rings over division rings.

Let [Z.sub.p], [Q.sub.p] and [C.sub.p] be denoted by the ring of

p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of [Q.sub.p] and let U D([Z.sub.p], [C.sub.p]) be denoted by the space of uniformly differentiable functions from [Z.sub.p] to [C.sub.p], cf.[2, 3, 5].

Let [[mu].sub.q] be the group of q-th roots of unity in the ring [Z.sub.p] of

p-adic integers. For each integer b with 0 [less than or equal to] b [less than or equal to] p - 1 and each

p-adic integer [alpha] [member of] [Z.sub.p] with [alpha] [equivalent to] 1 mod p, we put

(iv) A

p-adic integer r with [[upsilon].sub.p](r) = l can always be written as r = [p.sup.l]([e.sub.0] + [[summation].sup.[infinity].sub.j=1] [e.sub.j][p.sup.lj]) with [e.sub.0] [member of] [Z.sup.*.sub.p].