The additive group of the ring of p-adic numbers is not finitely generated

 1. The length of an element with respect to a (finite) set of generators

   Let G be a group and S be a generating set of G. For each g in G, the length of G with respect to S is the smallest non-negative integer k such that g can be written as s(1)s(2)...s(k), where s(i) is in S or s(i)^(-1) is in S for each i = 1, 2, ..., k. The length of the identity element is defined to be zero. In graph-theoretic language, the length of an element might be defined as the distance from the vertex representing that element to the vertex representing the identity element in the Cayley graph of G with respect to S. 

   If S is a finite generating set of an infinite group G, the number of elements of length not exceeding k is bounded above by (2|S| + 1)^k. Each such element should be written as s(1)s(2)...s(k) where for each i = 1, 2, ..., k, s(i) is in S or s(i)^(-1) is in S or s(i) is the identity element. Therefore, each s(i) can be chosen in (at most) 2|S| + 1 ways and there are at most (2|S| + 1)^k ways to choose s(1)s(2)...s(k). 

2. The additive group of the ring of p-adic numbers

   Let p be a prime number. An element of the ring of p-adic numbers can be written as a sequence (..., s(n), s(n - 1), ..., s(2), s(1)), where s(i) is an element of Z_(p^i) and the natural projection sends s(i + 1) to s(i) for each i. For example, if s(2) = 1 or s(2) = p + 1, then s(1) must be 1 since any number that is congruent to 1 or p + 1 modulo p^2 must be congruent to 1 modulo p.

   Assume that the additive group of the ring of p-adic numbers is generated by a finite set S. We will point out a contradiction by constructing an element that can't be written as an integral linear combination of elements of S. The idea comes from the notion of the length of an element with respect to a finite generating set. In more detail, we construct a sequence (..., s(n), s(n - 1), ..., s(2), s(1)) such that the length of s(n) with respect to the "generating set" given by the n-th coordinates of the elements of S becomes larger and larger as n becomes larger. However, the condition that the natural projection must sends s(i + 1) to s(i) requires some technical treatments. 

   The following fact is elementary/obvious: Given positive integers i < j and an integer a. A number is congruent to a modulo p^i if and only if it is congruent to one of the numbers a, a + p^i, a + 2*p^i, ..., a + (p^(j - i) - 1)*p^i modulo p^j. This means that for each fixed element of Z_{p^i}, there are p^(j - i) elements of Z_{p^j} that can be sent to that element via the natural projection. By using this fact, we might start with an arbitrary nonzero element s(1) of Z_p. Denote the length of s(1) with respect to the first coordinates of the elements of S by l(s(1)). To find suitable i and s(i) such that the length of s(i) with respect to the i-th coordinates of the elements of S is strictly larger than l(s(1)), we pick i such that p^(i - 1) is strictly larger than (2|S| + 1)^{l(s(1)) + 1}. Since there are p^(i - 1) elements s(i) that can be sent to s(1), but there are only at most (2|S| + 1)^{l(s(1))} elements of length l(s(1)), there exists one of these p^(i - 1) elements whose length is strictly larger than l(s(1)). By starting again with s(i) and repeating this process, we obtain the desired sequence (..., s(n), s(n - 1), ..., s(1)). 

   If (..., s(n), s(n - 1), ..., s(1)) can be written as an integral linear combination of elements of S, the length of each s(i) is bounded above by the sum of the absolute values of the coefficients in the integral linear combination, which is a contradiction. This means that the group of p-adic integers is not finitely generated. 

3. Some "exercises"

   Question 1: The construction above only tries to increase the length at specific/chosen points s(i). The fact that the length "becomes larger and larger as n becomes larger" is not fully considered. However, this can be proved by the increase of the length at specific points. Indeed, let (..., s(n), s(n - 1), ..., s(2), s(1)) be an arbitrary element of the ring of p-adic integers and S be a finite subset of the ring of p-adic integers. For each i, let l(s(i)) be the length of s(i) with respect to the set of the i-th coordinates of the elements of S. Prove that l(s(1)) <= l(s(2)) <= l(s(3)) <= ... 

   Question 2: Is there any other way to prove that the additive group of the ring of p-adic numbers is not finitely generated? 

   Question 3: Can the same approach be used to prove that the multiplicative group of invertible elements of the ring of p-adic numbers is not finitely generated? If the answer is Yes, is there any generalized version of this idea (for example, by considering the inverse limits of groups)?