2010 Spring MATH 274-01
Bulletin Course Description Binary quadratic forms; orders, integral closure; Dedekind domains; fractional ideals; spectra of rings; Minkowski theory; fundamental finiteness theorems; valuations; ramification; zeta functions; density of primes in arithmetic progressions. Instructor: Staff
(Instructor named in bulletin description above may not be current. For current instructor, see listing below.)
Title NUMBER THEORY Department MATH Course Number 2010 Spring 274 Section Number 01 Primary Instructor Saper,Leslie D Prerequisites Prerequisites: Mathematics 201 or 251 or consent of instructor. Course Homepage www.math.duke.edu/faculty/saper/Instruction/math274.S10/
Synopsis of course content
This course is an introduction to modern algebraic number theory. We will cover the basics of number fields and rings of integers, Dedekind domains, valuations and completions. We will then discuss the decomposition of primes in extension fields and ramification theory. We will prove the basic theorems in the geometry of numbers, the Dirichlet unit theorem and the finiteness of the class group. There will be many examples and applications to the study of quadratic forms, low degree extensions, and cyclotomic fields. Depending on time and interest, we will discuss other topics such as the analogy between number theory and algebraic geometry in dimension 1, adeles and ideles, and L-functions.
Prerequisites
Math 251 or permission of instructor. Specifically the student should be familiar with group theory, commutative rings and modules over them, classification of finitely generated modules over a principal ideal domain, and field theory including extensions of fields, structure of finite fields, and Galois theory. Students who are uncertain about the prerequisites should consult the instructor.
Textbooks
The following references will be put on reserve; the first will be available in the bookstore:
- "Algebraic Number Theory" by Lang
- "Algebraic Number Theory" by Neukirch
- "Number Theory" by Borevich and Shafarevich
Additional Information
The course should be of interest to graduate students or undergraduates interested in algebra, algebraic geometry, differential geometry, mathematical physics, topology or another area of mathematics which interacts with number theory.
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