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Algebraic Number Theory (Math 607)


Here is the course description: This two-term algebraic number theory sequence will introduce standard topics essential for students considering working in number theory and also potentially useful for students working in related fields, including algebraic geometry, representation theory, and topology.  These courses will focus on the structure of number fields (finite extensions of the rational numbers), including ideal class groups, unit groups, cyclotomic extensions, quadratic reciprocity, special cases of Fermat’s Last Theorem, local fields, global fields, adeles, and an introduction to class field theory.  The main prerequisite is abstract algebra at the level of the 500-level algebra sequence (with an emphasis on the Galois theory covered in Math 546), as well as elementary knowledge of modules.  Students who have seen some commutative algebra (at the level covered in the 600-level algebra sequence) will be better prepared.  Students who have not seen commutative algebra at the level of 600-level algebra will need to do a little extra work or accept a few facts as black boxes.  Topics and pace may be adjusted according to the background and interests of the students enrolled in the course (e.g. since some of these topics have been covered in the student number theory seminar in 2017-2018).