Quadratic Reciprocity

In this chapter, we consider integers that are congruent, modulo a prime p, to the square of another integer. This is equivalent to considering when an element of https://www.w3.org/1998/Math/MathML"> ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003318712/0a8d6242-fb38-4fc0-af6c-0a4803a1cbee/content/C006_equ_0001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> _p can be written as the square of another element of https://www.w3.org/1998/Math/MathML"> ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003318712/0a8d6242-fb38-4fc0-af6c-0a4803a1cbee/content/C006_equ_0002.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> _p. For example, in https://www.w3.org/1998/Math/MathML"> ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003318712/0a8d6242-fb38-4fc0-af6c-0a4803a1cbee/content/C006_equ_0003.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ₁₁, [3] is a square, because [3] = [5]². Our ultimate objective is a discussion of one of the most famous theorems of elementary number theory, the Law of Quadratic Reciprocity.