DescriptionOne of the surprising gems that one encounters in elementary number theory is the quadratic reciprocity law discovered by Legendre, one case of which being, for primes p and q congruent 1 mod 4, that p is a square mod q if and only if q is a square mod p (e.g. 17 is clearly a square mod 13, and both =1 mod 4, so we can be sure that 13 is a square mod 17 (can you find one?)).Gauss was the first to give a complete proof in the early 18th century, and in fact, in the course of his later life he found several completely different further ones, and they are still worth studying today. He also laid the groundwork for an answer to a similar statement for cubes, and even higher powers modulo prime ideals (in each case one needs to pass to a larger field in order to get a suitable formulation). The first proofs of the cubic and quartic reciprocity laws were subsequently found by Eisenstein (age 21). These amazing relationships are already a very rich albeit demanding playground for an ambitious project. Background in Number Theory would be highly recommended, as the best formulation will involve the decomposition behaviour of primes when viewed in a number field as a product of prime ideals. A high-brow conceptual generalisation was found, in the context of Class Field Theory, by Artin which roughly relates, for number fields with abelian Galois group, a given ideal class with its `reciprocal' counterpart, a specific Galois automorphism. One suggestion for the project is to understand, formulate and corroborate the reciprocity laws, starting from the quadratic one, but with novel and highly non-trivial examples--Pari/GP is a free computer algebra package with fantastic tools for number theory enthusiasts. But of course one can also take a purely theoretical route. A beautiful book to get started with related questions is "Primes of the form x^2+ n y^2" by David Cox. A treasure trove of interrelationships among the many discoveries in this, with plenty of historical context, is the book by Lemmermeyer. PrerequisitesFamiliarity with Algebra, as well as Elementary Number Theory, is strongly recommended.ResourcesDavid Cox: Primes of the form x^2+ny^2. Beautiful introduction to the surprisingly structured way in which primes conspire to be expressible as a specialisation of some specific quadratic form.
Kenneth Ireland, Michael Rosen: A classical introduction to modern number theory A well known textbook. Chapters 5, 9, 13, 14 are particularly relevant. Franz Lemmermeyer: Reciprocity Laws. A kind of encyclopaedic treaty of the statements and history of the many different reciprocity laws. Bostwick Wyman: What is a reciprocity law? An outlook into an even more general picture, including Artin reciprocity. |