Research and Publications
My Ph.D research was on the classical approach to
Fermat's last theorem using cyclotomic field theory. My thesis advisor
Prof. Dinakar Ramakrishnan .
These days I am working on Artin's Primitive Roots Conjecture.
My latest preprint, to appear in the Hardy Ramanujan Journal (this work is not mentioned in talk below).
Notes from a Graduate Student Seminar talk of mine on Artin's Conjecture
This came as a byproduct of my interest in the asymptotics of the quantities associated to the cyclotomic field. For instance, it has not been proven that there are infinitely many p-th cyclotomic fields K_p where the prime p does not divide the class number of K_p.
- Mean Values of Certain Multiplicative Functions and Artin's Primitive Roots Conjecture Hardy-Ramanujan Journal, vol. 37, February 2015.
- Note on a Fermat type diophantine equation [Journal of Number Theory, vol. 99, 29-35, 2003]
- On a Fermat type diophantine equation [Journal of Number Theory, vol. 80, 174-186, March 2000]
- Vandiver Revisited Journal of Number Theory, vol 57,1996
- History of Waring's Problem, Mathematics Student (A journal of the Indian Math. Soc.),vol.61, 1992
Resources for Number Theory
Several resources are available for number theorists, including lists of number theorists,
preprint servers, archives, and computational resources such as PARI and the elliptic
curve calculator. All these and more can be found on the
Number theory web.
( link to the site in Brisbane, Australia. There are other mirror sites available).
Fermat's Last Theorem and Cyclotomic fields
Starting with Kummer, a long list of mathematicians including Mirimanoff, Pollaczek,
Furtwangler, Morishima, Eichler, Inkeri, and Vandiver tried to use algebraic number theory , and the theory of cyclotomic fields in particular, to solve Fermat's last theorem. They met with
a great deal of success, especially on what is known as the First case, as chronicled in
P. Ribenboim's "Thirteen lectures on Fermat's Last Theorem." Ultimately, though, the theory
of elliptic curves and modular forms in what is now called arithmetic geometry proved
more powerful, and Andrew Wiles was able to prove the "theorem" in 1995 in its entirety.
( Here is an article by
G. Faltings on Wiles' proof. For a detailed list of resources available on
Fermat's last theorem, go to
Eric Weisstein's treasure trove on Fermat's last theorem.
Notes from a lecture of mine on this topic
But the work on cyclotomic fields has by itself yielded many fundamental results that
are useful in many branches of number theory and arithmetic geometry quite independent
of their relevance to Fermat's last theorem, as can be seen from L. Washington's
"Introduction to cyclotomic fields" or S. Lang's "Cyclotomic fields."