Math 254: Introduction to Algebraic Number Theory
Instructor: Chad Schoen
Prerequisites:
Math 251 or strong performance in Math 200 or the equivalent. Students
will be expected to enter the course with a working knowledge of the
following basic notions in algebra: Groups, actions of groups on a set,
commutative rings, ideals, modules over commutative rings, principal
ideal domains, unique factorization domains, the classification of
finitely generated modules over principal ideal domains, field
extensions, the structure of finite fields, basic Galois theory.
Intended audience:
The course should be helpful to graduate students considering working
in algebra, algebraic geometry or another area of mathematics which
interacts with number theory. It should be of interest to advanced
undergraduates who have fulfilled the prerequisites and who are interested in
learning more about algebraic number theory.
Text:
Neukirch, J.; Algebraic Number Theory
Topics:
Quadratic equations in two variables, Number fields and function
fields, orders in number fields and algebraic curves over finite
fields, affine schemes, integral closure and desingularization,
Dedekind domains, the class group, applications to binary quadratic
forms and diophantine equations, decomposition of primes in extension
fields, valuation theory, ramification theory, the geometry of numbers,
finiteness of the class group and Dirichlet's unit theorem, extensions
of global fields with fixed ramification, bounds on discriminants.
Relationship with algebraic geometry:
Students who enter the course with a background in algebraic geometry
will have the opportunity to deepen their knowledge by working special
problems. However algebraic geometry is not a prerequisite and students
who have had no exposure to algebraic geometry will not be at a
disadvantage.
Additional references:
Lang, Algebraic Number Theory
Samuel, Introduction to the Algebraic Theory of Numbers
Serre, Local fields