MTH 617 Algebraic number theory, 3-0-0-4, Summer semester 2002


Algebraic number theory is the branch of number theory, that employs abstract algebraic techniques for investigating properties of (rational) integers. It developed in the nineteenth and twentieth centuries during attempts to prove the so-called Fermat's last theorem (FLT) which states that there are no solutions in positive integers x,y,z of the equation x^n+y^n=z^n, where n is an integer larger than 2. FLT has been finally proved by Andrew Wiles, but several other problems and conjectures continue to remain unsettled.

It is difficult, if not impossible, to cover all the facets of algebraic number theory in a one-semester course. I would like to introduce some simple (?) properties of number fields (finite (and hence algebraic) extensions of the field of rational numbers) and their rings of integers. If time permits, some elementary p-adic methods and some analytic tools will also be discussed. This course is meant for advanced undergraduate and beginning graduate students in the discipline of mathematics. Computer scientists and engineers interested in primality proving, integer factorization and cryptology should also find this course useful. My focus will be on discussing known results rather than on concentrating on the research topics in this area. Research students (in mathematics and computer science) lacking proper background in this subject may take this course as a starting point in their research careers.

Interested students may contact me by e-mail or meet me in my office (Faculty building, Room No 567).

Prerequisites for registering


1. Algebra preliminaries [pdf] [ps]
Unique factorization domains, principal ideal domains and Euclidean domains, ideal arithmetic in rings, modules and algebras, finite fields.
2. Commutative algebra [pdf] [ps]
Localization, quotient fields of integral domains, integral dependence, normal domains, chain conditions, Noetherian rings and modules.
3. Number fields and number rings [pdf] [ps]
Algebraic numbers and integers, number fields, complex embeddings, number rings, integral bases.
4. Unique factorization of ideals [pdf] [ps]
Dedekind domains, fractional ideals, unique factorization, factorization of rational primes, inertia, ramification and splitting, norms and traces.
5. Class group theory (Notes included in Chapter 4)
Class groups and class numbers, finiteness of class numbers, computation of class numbers for quadratic fields.
6. Structure of the group of units in a number ring [pdf] [ps]
Lattices, Minkowski's convex-body theorem, Dirichlet's unit theorem.
7. Case studies (distributed among the previous topics)
Quadratic and cyclotomic number fields.
8. p-adic numbers [pdf] [ps]
p-adic integers and p-adic numbers, p-adic norms, Ostrowski's theorem, completeness, Hensel's lemma.
9. Applications (Not covered for lack of time)
Primality proving and integer factorization, the number field sieve.
10. Analytic tools (Not covered for lack of time)
Characters of finite Abelian groups, Dirichlet series, Dirichlet's theorem on primes in arithmetic progression.

Note: You can download the course notes by clicking on the above links. The documents are proof-checked, but the possibilities of unnoticed errors can not be ruled out. If you locate errors/inconsistencies/... or merely want to pass a comment, write to me. Particularly for the students, try to solve as many exercises appearing in the notes as possible.

Test schedule

SyllabusQuestion paper
Mid-semester test 1September 8, 200225 1 hourTopic 1 [Questions] [Solutions]
Mid-semester test 2October 22, 200230 1 hourTopics 2,3 [Questions] [Solutions]
End-semester testNovember end / December beginning 45 3 hoursTopics 1--7 [Questions] [Solutions]
(All tests were open-note and open-time.)


Books on algebraic number theory demand varying prerequisites on the part of the readers. I found the following books suitable for this course. The following books may be consulted for relevant algebraic tools. The following books are somewhat advanced, but can still be followed: