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

### Announcements

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

• Knowledge of algebra and linear algebra (groups, rings, fields and vector spaces) is required.
• Knowledge of elementary number theory is strongly recommended. Deficient students may read this note (not proofchecked yet).
• No knowledge of analytic number theory is required.
• No knowledge of commutative algebra is required.

### Syllabus

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)
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

TestTimeTotal
points
Duration
(strechable)
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.)

### References

Books on algebraic number theory demand varying prerequisites on the part of the readers. I found the following books suitable for this course.
• Jody Esmonde and M Ram Murty, Problems in algebraic number theory, GTM #190, Springer-Verlag, 1999. [Locally available]
• Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, GTM #84, Springer-Verlag, 1990. [Locally available]
• Richard A Mollin, Algebraic number theory, CRC Press Series on Discrete Mathematics and Its Applications, Chapman & Hall, 2000. [A copy available with me.]
• Raghavan Narasimhan, A Raghavan, S S Rangachari and Sunder Lal, Algebraic number theory, School of Mathematics, Tata Institute of Fundamental Research, Bombay, 1966. [Locally available]
• Harry Pollard and Harold G Diamond, The theory of algebraic numbers, Dover Publications, 1998. [A copy available with me.]
The following books may be consulted for relevant algebraic tools.
• W A Adkins and S H Weintraub, Algebra: An approach via module theory, GTM #136, Springer-Verlag, 1992. (for algebra preliminaries) [A copy available with me.]
• M F Atiyah and I G MacDonald, Introduction to commutative algebra, Addison-Wesley, 1969. (for commutative algebra) [A copy available with me. I didn't check, but believe that this book is also locally available.]
The following books are somewhat advanced, but can still be followed:
• Gerald J Janusz, Algebraic number fields, American Mathematical Society, 1995. [Locally available]
• Serge Lang, Algebraic Number Theory, GTM #110, Springer-Verlag, 1994. [Locally available]
• Jürgen Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, #322, Springer-Verlag, 1999. [Locally available]
• Paulo Ribenboim, Classical theory of algebraic numbers, Universitext, Springer, 2001. [An older edition is locally available]