References: The "official" textbook for this
course (not required buying it) is: Frohlich and Taylor Algebraic
Number-Theory.
The following are some books related to the course material:
S. Lang, Algebraic Number Theory , Springer-Verlag, 1994.
G. J. Janusz, Algebraic Number Fields , Providence RI, A.M.S. 1996.
J. Neukirch, Algebraic Number Theory , Springer-Verlag, 1999.
P. Samuel, Algebraic Theory of Numbers
Outline of the course: This course is a one-semester,
introductory graduate-level course in algebraic number theory. Topics expected to be covered
include: rings of integers, Dedekind domains, number fields, the ideal class
group and units, cyclotomic fields, valuations, local fields, adeles
and ideles.
Course Description: Algebraic Number Theory begins
with the goal to discover to what extent the Fundamental Theorem of
arithmetic (on factorization of an integer by a product of prime
numbers) holds, or fails to hold, in finite field extensions of the
rationals (number fields). This course covers the basic structure of
these fields and it focusses on the study of algebraic
integers in number fields which play the role analogous to that of the rational
integers in the field of the rationals. This course is supposed to
give the prerequisites for a more advanced course on class field
theory. It is an essential course for anyone wishing to pursue
graduate studies in number theory and arithmetic geometry.
Prerequisites: Abstract algebra: including groups,
rings and ideals, fields and Galois theory, e.g. 110.401-402 (or equivalent).
Special Notice: This course is listed as a
graduate-level course and will be taught as such even in the presence
of undergraduate students or graduate students in other subjects i.e.
without a full undergraduate math major. That means I will
expect a level of scholarly and mathematical maturity appropriate to a
first-year graduate student in mathematics. I also expect
to complement some of the topics with further material. For this reason I warmly suggest
ALL STUDENTS ENROLLED to take notes in class.
Grading: Homework will be assigned periodically
during lectures. The exercises will be collected by the instructor
who will assign an overall grade. The final grade will
be determined by two components: 1) active participation at lectures
time, 2) homework performance.