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Chapter 17 Quadratic Reciprocity

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 16.5.2 directly for every single one.
It turns out that finding out when numbers have square roots (mod \(p\)) is not hopeless – quite the opposite is true! After raising our spirits with some simple but powerful observations, we will make our way to the great theorem that is the title of this chapter. Using it, we will derive almost effortlessly results regarding quadratic residues that originally took a great deal of work.