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 (which 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 formul ation 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 cla ss with its `reciprocal' counterpart, a specific Galois automorphism. Class Field Theory is widely seen as one of the triumphs of early 20th century Number Theory. The statements are stunningly beautiful but, as often in arithmetic, the proofs are rather hard. One suggestion for the project is to understand, formulate and corroborate the main claims, but with new and highly non-trivial examples--Pari/GP is a free computer algebra package with fantastic tools for number theory enthusiasts. Of course one can also take a purely theoretical route. A treatment of Artin's reciprocity law with novel examples would constitute a very demanding but equally satisfying highlight. PrerequisitesAlgebra, Number Theory, and ideally also Galois Theory.ResourcesDennis Garbanati: Class Field Theory summarized. (Online) A neat introduction to the subject. Nancy Childress: Class Field Theory. An introductory textbook. Juergen Neukirch: Class Field Theory. A classic. David 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. Gerald Janusz: Algebraic Number Fields. The book is directed toward students with a minimal background who want to learn class field theory for number fields. The only prerequisite for reading it is some elementary Galois theory.
Kenneth Ireland, Michael Rosen: A classical introduction to modern number theory A well known textbook. Chapters 5, 9, 13, 14 give reciprocity laws. Franz Lemmermeyer: Reciprocity Laws. A treasure trove of interrelationships among the many discoveries in this, with plenty of historical context (kind of encyclopaedic). Bostwick Wyman: What is a reciprocity law? An outlook into an even more general picture, including Artin reciprocity. |