Of course, there are infinitely many primes, but what if we decided that we only wanted to have one? That's what happens when we work locally with a chosen prime p. In this talk I'll introduce this idea by constructing the p-adic numbers, Q_p, two ways: analytically and algebraically. I'll also talk about some of the fun properties of Q_p. Spoiler: have you ever wished that series converged if and only if their terms went to zero? Then I have the field for you! To wrap things up, I'll briefly talk about my research, which takes place in a local setting. 

While we love p-adics, there is an obvious question when working p-adically: which prime p should we use? Different problems necessitate different answers to this question, but my personal favorite answer is to not discriminate and just consider all the primes simultaneously. This leads to the ring of adeles (over the rational numbers), which is an easily described subspace of the product of Q_{p} over each prime p (including p = infty!). In this talk, we will introduce adeles and their multiplicative analog, ideles, and discuss some basic results. We will also discuss how adeles became a cornerstone of modern number theory by highlighting some classical results from class field theory. We will finish by discussing how adeles are used today by discussing adelic points of algebraic groups.