Title of the course: Prime Factorization in Algebraic Number Fields
Instructor: E. Mehmet Kıral
Institution: RIKEN AIP (Tokyo)
Dates: 17-23 July 2023
Prerequisites: Linear Algebra, familiarity with basic mathematical structures.
Level: Graduate, advanced undergraduate, beginning undergraduate
Abstract: We take the number field K, a finite algebraic extension of the rational numbers as our main object of study and use the interplay between linear algebra (the field is a finite dimensional vector space over Q) and the multiplication in the field in exploring the structures in the field itself. One of thses structures are full Z-lattices ın K, and their ring of coefficients which also end up being full Z-lattices. The ring of integers of the field sits in a distinguished position among all these orders and is the only such ring inside the number field where unique factorization into ideals holds true.
The course doesn’t technically require much, however the focus will NOT be on simply getting the facts out there, but rather understanding—to the best of our ability—why certain objects are special, and thinking deeply about exactly which interplay accounts for the observed number theoretic phenomena. As such I could suggest the course also for those who already have studied the topics once, and would like to allow the time to think again on these beautiful objects. A kind of “polish”, if you will.
Language: TR, EN