Meghan De Witt and Dr. Darrin Doud, Mathematics Department
My project was to finish a thirteen year old unsolved problem that provided an important computational example of a conjecture in the field of Algebraic Number Theory. Originally, we worked with the techniques of elliptic curves and class field theory to try and isolate the objects that we were searching for, this involved using a complex computer search as well as advanced computer algebra systems. This allowed us to complete the work to the level that we had earlier anticipated. However, we then continued with the project and were able to progress further on the problem and actually prove the correctness of our work. This was originally believed to be impossible. We began by using a computer algebra system to compute certain values associated to the objects, known as fields, that we wished to find. We then took these values and created a computer search to find a set of equations associated with these fields. Using advanced techniques we were then able to compute the equation defining each of the fields we were looking for. We showed that these were only the possible objects satisfying the conditions we were looking for, and since we knew such objects existed it therefore followed that we had in fact found the precise fields desired. These now finished examples provide important new insight into research relating to Serre’s Conjecture and its generalization.
This work led to an interest in many areas of mathematics which I am now exploring in graduate school at the University of Wisconsin in Madison. These include advanced number theory, class field theory, Kummer theory, and the theory of elliptic curves.
I have presented this work at an American Mathematical Society conference, as part of the spring research conference of the College of Physical and Mathematical Sciences, and to Young Mathematicians Conference. I have also submitted a paper based on the results of this research for publication in the International Journal of Number Theory; it is currently under review.