It is likely that divisibility is the first research topic in number theory and it had been settled and developed by Gauss based on his famous theory of congruences. An integral and important part of this theory, congruences, is devoted to the binomial congruences. The essence of the binomial congruences is the theory of power residues whose cornerstone is “the reciprocity laws”. Although discovering and stating the contents of these laws was relatively easy, however finding their proofs was a difficult task. For more than two centuries many mathematicians, Gauss himself on the top of them introduced many new methods, and presented many proofs that are connected to different areas of mathematics that seems to have no connection with number theory. The first form of these laws is the Quadratic Reciprocity Law, QRL. The first step toward the proofs was about the Quadratic Reciprocity Law, but the principles found in the different proofs of the QRL could be generalized in order to drive the General Reciprocity Law. After C. G. Bachet de Méziriac [Problèmes plaisants et délectables qui se font par les nombres. 1st ed. (1612); 2nd ed. (1624)] wrapped the theory of linear Diophantine equations up, mathematicians were faced with the solution of the equations of the second degree, in particular the binomial congruence of degree two. to be more specify, let \(p\) and \(q\) be two odd prime integers. They wanted to find simple conditions for the solvability of the following congruence \[ x^2\equiv p \mod{q}. \] Legendre used Fermat’s little Theorem to obtain his results. In fact it follows from \[ x^2\equiv p\mod{q}\text{ and }p^{q-1}\equiv1\mod{q} \] that the solvability of the \(x^2\equiv p \mod{q}\) depends on \(p^{((q-1)/2)}\). In fact, if \(p^{((q-1)/2)}\equiv 1\mod{q}\), then \(x^s\equiv p \mod{q}\) is solvable; if, however, \(p^{((q-1)/2)}\equiv-1\mod{q}\), then the congruence is not solvable. Legendre calls this law the Quadratic Reciprocity Law, QRL. According to the above observations, Legendre adopted the following great notation \[ ((a/p))=\begin{cases} 1, &\text{if }x^2\equiv a \mod{p}\text{ is solvable}\\ -1, &\text{if }x^2\equiv a \mod{p}\text{ is not solvable.}\end{cases} \] According to this notation, the Quadratic Reciprocity Law can be presented as follows.
Theorem. Let \(p\) and \(q\) be two odd prime integers. Then

(a)

\((((-1)/p))=(-1)^{((p-1)/2)}\).

(b)

\(((2/p))=(-1)^{((p^2-1)/8)}\).

(c)

\(((p/q))((q/p))=(-1)^{(((p-1)/2))(((q-1)/2))}\).

The book under review contains a translation of Oswald Baumgart’s thesis titled, “On the Quadratic Reciprocity Law, a comprehensive presentation of its proofs” [see JFM 17.0026.02]. Oswald Baumgart’s thesis was written in Göttingen a hundred years after Legendre’s publication of the Quadratic Reciprocity Law was known to him in 1885. The translator provides a treasure of classical proofs of the Quadratic Reciprocity Law, that are based on different methods from different areas of mathematics such as complex analysis, cyclotomic fields, quadratic forms, and so on. In almost all proofs one can see a shade of Gauss’ ideas. One of the main keys in most of the proofs is Gauss’ lemma. Consider \(S=\{-((q-1)/2),-((q-3)/2),\dots,-1,1,2,\dots,((q-1)/2)\}\). This is called the set of least residues \(\mod{q}\). Let \(\mu\) be the number of negative least residues of the integers \(p,2p,3p,\dots(((q-1)/2))p\). Then
Lemma (Gauss’ lemma). \(((p/q))=(-1)^{\mu}\).
Among all other proofs, the analytic proof by Eisenstein receives a smaller discussion. The book has an excellent comparative discussion of many proofs along with historic notes and comments by translator. It contains a vast list of references that are updated. A full list of all proofs up to 2014 of the “Quadratic Reciprocity Law” are included. This excellent book is a necessary one for any number theorist. Every student in the field can find a lot of virgin ideas for further research as well. This book should be a good resource for mathematics historian as well.