Reciprocity Laws: From Euler to Eisenstein

 We typically learn (and teach!) the law of quadratic reciprocity in courses on Elementary Number Theory. In that context, it seems like something of a miracle. Why should the question of whether p is a square modulo q have any relation to the question of whether q is a square modulo p? After all, the modulo p world and the modulo q world seem completely independent of each other. (Isn't that what the Chinese Remainder Theorem says?) The proofs in the elementary textbooks don't help much. They prove the theorem all right, but they do not really tell us why the theorem is true. So it all seems rather mysterious. On top of that, the books often tell us that Gauss gave a whole bunch of proofs of this theorem, and that hundreds of proofs are now known... and we are left with a feeling that we are missing something. What we are missing is what Franz Lemmermeyer's book is about. Right at the beginning, he makes the point that even the quadratic reciprocity law should be understood in terms of algebraic number theory, and from then on he leads us on a wild ride through some very deep mathematics indeed as he surveys the attempts to understand and to extend the reciprocity law. Most of the book deals with the many "higher reciprocity laws" which were a central theme in nineteenth century number theory. As the introduction suggests, in the twentieth century this theme developed into what is now known as "Class Field Theory," and the only unfortunate thing about this book is that it doesn't follow the thread all the way to the end. But never fear: the author promises us a second volume, to pick up where this one leaves off and lead us all the way to the Artin Reciprocity Law and (I hope!) beyond. For now, this is a very good expository account of some difficult, deep, and beautiful mathematics.

Fernando Gouvêa (fqgouvea@colby.edu) is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.