Its inverse is given by the coherent sequence where x k satisfies for . We have where for . The rational numbers r which have this form are just the ones for which , since . These are called p -integral rational numbers. Corollary 5.3 R p is a ring, with , and .
Every rational number is in for all but a finite number of primes p .
Every nonzero element may be uniquely expressed as where , and . For , we have if and only if . Thus every p -adic integer is either a unit or a multiple of p , but not both. For nonzero we define , where is the power of p appearing in the factorization above. Thus m is the largest integer for which .