Profinite Groups

γ = c0 + c1p + c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it must have a solution modulo p for all n: to prove it does not have a solution, therefore, it suffices to show that it does not have a solution in Zp for some prime p. We can express the expansion of elements in Zp as Zp = lim ←− n Z/pZ