So far, we have determined at least when some quadratic congruences have solutions, but at the pace set thus far, most cases should seem beyond reach. We certainly won't want to use Theorem 16.5.1 directly for every single one.

Yet one might think we must, and that our task of finding out when numbers have square roots (mod \(p\)) is hopeless. But quite the opposite is true! In fact, we will derive results that took quite a bit of work almost effortlessly using the great theorem that is the title of this chapter.