Quadratic reciprocity in the context of "Carl Friedrich Gauss"

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form ${\displaystyle x^{2}\equiv a{\pmod {p}}}$ for an odd prime ${\displaystyle p}$; that is, to determine the "perfect squares" modulo ${\displaystyle p}$. However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case ${\displaystyle p\equiv 3{\pmod {4}}}$ using Euler's criterion one can give an explicit formula for the "square roots" modulo ${\displaystyle p}$ of a quadratic residue ${\displaystyle a}$, namely,