Introduction

Historically, thanks to Gauss, the quadratic reciprocity law marked the beginning of algebraic number theory. Therefore it it deserves a good dose of attention. However, whacking the definition to the beginner would not work pretty well.

We consider the equation

$$x^2+by = a,\; (a,b\in \mathbb{Z}),$$

one of the simplest non-trivial multi-variable Diophantine equations that can be imagined. Trying to violently search all solutions without any precaution is not wise. Therefore we consider reductions first. In order that $x^2+by=a$ has a solution, it is necessary that

$$x^2 \equiv a \pmod{b}.$$

Then the Chinese remainder theorem inspires us to first look into the case when $b$ is a prime. The case when $b=2$ is excluded because we are only allowed to study whether $x$ is odd or even.

Therefore we study the equation $x^2=a$ in the finite field of order $p$ where $p \ne 2$. We give a very straightforward characterisation, which is seemingly stupid. For $a \in \mathbf{F}_p^\ast$, define

$$\left(\frac{a}{p}\right)=\begin{cases}\;\;\;1 & \text{$x^2=a$ has at least one solution in $\mathbf{F}_p$,} \\-1 & \text{$x^2=a$ has no solution in $\mathbf{F}_p$.}\end{cases}$$

It is also convenient to define $\left(\frac{0}{p}\right)=0$.

This post will start with an equivalent form that is easier to compute (although less intuitive)....