Through the simple question, which prime numbers can be written in the form x² + n y² for a natural number n, and integers x and y, one is led rather quickly to the realm of algebraic number theory and class field theory. We will first discuss classical results by Fermat, Euler, and Gauss (including Gauss composition of binary forms and genus theory). Then we will review some algebraic number theory and start our journey towards understanding the question in the title through class field theory for quadratic extensions of the rationals.
Every student is assumed to give one talk in the first half and one talk in the second half of the semester, each of 45 minutes. A short explanation on how to prepare for a talk in this seminar can be found here. Typically the speakers for week n will be determined at the end of the seminar in week n-2. However, note that since we want to follow the book of Cox' without leaving any gaps, the speakers of week n will continue where the speakers of week n-1 left off. The exact amount of topics to be treated is thus only available in week n-1, but one can start reading before that.
As a preparation for your talk, we offer you a non-mandatory meeting with one of us in the week before your talk. We are of course also willing to answer questions by email as well.
|19.02.19||Menny and Andreas||Introduction, organizational matters and quadratic reciprocity.|
|26.02.19||Roman and Thomas||More on quadratic reciprocity (quadratic characters). Quadratic forms, equivalence classes and reduced forms.|
|05.03.19||Moritz and Nadine||Finiteness of the number of reduced forms (the class number), class number one property.|
|12.03.19||Ajith and Emie||Elementary genus theory, definition of the class group.|
|19.03.19||Carlo and Felix||Elements of the principal genus and squaring in the class group.|
|02.04.19||Carlo and Shengxuan||A short outlook on cubic and biquadratic reciprocity. Number fields, ring of integers, unique factorization of ideals in Dedekind domains.|
|09.04.19||Emie and Thomas||tbd|
Solving exercises is an important part of this seminar and a prerequisite for every attending student to obtain the credits. When you have solved an exercise, post it on this overleaf, so that we and other students can have a look at it. We expect every student to solve and post at least 4 exercises during the semester which should not on the topics presented by the student and which have yet not been solved by another student. Amongst these 4 exercises at least two should be from Chapter two of Cox's book. Note also that some exercises in the book are much simpler than others and you will not be allowed to solve only such exercises. You may of course suggest other exercises of your interest.
The main reference is Primes of the form x^2+ ny^2 (Fermat, class field theory, and complex multiplication) by David A. Cox (Vol. 34. John Wiley & Sons, 2011). Further useful references include: