DOI link for Quadratic Residues
Quadratic Residues book
As with everything else, so with mathematical theory, beauty can be perceived, but not explained.
Arthur Cayley (1821-1895), British mathematician
We have encountered the notion in the title of this chapter already. For instance, when we introduced power residues in Definition 3.4 on page 155, quadratic residues were given as an illustration. Also, in Examples 2.8-2.12 in Chapter 2, quadratic congruences were illustrated. Indeed, quadratic congruences are the next simplest after linear congruences the solutions for which we classified in Theorem 2.3 on page 84. As we have seen, it is from quadratic congruences that the notion of quadratic residues arise. For instance, such questions arise as: given a prime p and an integer a, when does there exist an integer x such that x2 ≡ a(mod p)? Such queries were studied by Euler, Gauss, and Legendre, the latter having his name attached to the symbol that we study in §4.1, a symbol which provides a mechanism for answering the above question.