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
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
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
- No knowledge of analytic number theory is required.
- No knowledge of commutative algebra is required.
- 1. Algebra preliminaries
- Unique factorization domains, principal ideal domains and Euclidean
domains, ideal arithmetic in rings, modules and algebras, finite fields.
- 2. Commutative algebra
- Localization, quotient fields of integral domains, integral dependence,
normal domains, chain conditions, Noetherian
rings and modules.
- 3. Number fields and number rings
- Algebraic numbers and integers, number fields, complex embeddings,
number rings, integral bases.
- 4. Unique factorization of ideals
- 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
- 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
- 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.
(All tests were open-note and open-time.)
|Mid-semester test 1||September 8, 2002||25
||1 hour||Topic 1
|Mid-semester test 2||October 22, 2002||30
||1 hour||Topics 2,3
|End-semester test||November end / December beginning
||3 hours||Topics 1--7
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.
- Jody Esmonde and M Ram Murty, Problems in algebraic number theory,
GTM #190, Springer-Verlag, 1999.
- Kenneth Ireland and Michael Rosen, A classical introduction to
modern number theory, GTM #84, Springer-Verlag, 1990.
- 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.
- Harry Pollard and Harold G Diamond, The theory of algebraic numbers,
Dover Publications, 1998.
[A copy available with me.]
The following books are somewhat advanced, but can still be followed:
- 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.]
- Gerald J Janusz, Algebraic number fields, American Mathematical
- Serge Lang, Algebraic Number Theory, GTM #110, Springer-Verlag,
- Jürgen Neukirch, Algebraic Number Theory, Grundlehren der
mathematischen Wissenschaften, #322, Springer-Verlag, 1999.
- Paulo Ribenboim, Classical theory of algebraic numbers,
Universitext, Springer, 2001.
[An older edition is locally