Applications of the p-adics As I was preparing to spend a month studying p-adic analysis, I realized that I've never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field $\mathbb Q_p$ has many nice properties, e.g. $X\subset Q_p$ closed and bounded implies $X$ compact, $\sum\limits_{n=0}^\infty x_n$ exists iff $\lim\limits_{n\to\infty}x_n=0$, its ring of integers is compact, and any open set is a union of disjoint open balls. But how are properties of $\mathbb Q_p$ translated to applications to other mathematical problems? Are there any significant mathematical problems which have been resolved by applying the theory of p-adic numbers?
 A: The Weil Conjectures are proved using $p$-adic numbers ($\ell$-adic cohomology); but the statement of these theorems does not make use of $p$-adic numbers.  These are very important theorems in number theory and arithmetic geometry that explain the pattern you find when you count the number of points on an algebraic variety over a finite field and its extensions.  
The generating function built from these point counts is a rational function, called a local zeta function.  The Weil Conjectures give us an analogue of the Riemann Hypothesis: if we have the local zeta function of a curve, we know that all of its zeros lie on the critical line!
A: This is almost like asking "what are real numbers good for". It would require several tomes of terse enumeration of key phrases to compile a comprehensive list.
Here is a random collection of terms for you to google (all these are mainly number theoretic, since that's where I come from, but there are plenty of applications in other areas; for each of these terms, it would require several months to several years to appreciate its depth): Hasse principle, local class field theory, $l$-adic Galois representations, Iwasawa theory and $p$-adic $L$-functions, rigid analytic geometry. I had better stop here. 
