Project Title: Diophantine equations over quadratic fields

Adviser: Griff Elder

Description: In 1934, A. Aigner (an Austrian mathematician) solved the equation x^4+y^4=z^4 over quadratic extensions of the rationals. Since then both D. K. Faddeev (a famous Russian mathematician) and L. J. Mordell (a famous English mathematician) have considered the question. During 2002/2003 academic year, UNO junior Eric Manley extended Aigner's result to the equation x^4+p y^4=z^4 for p a prime congruent to 3 modulo 8. His result appeared in the Rocky Mountain Journal of Mathematics in 2006. We are interested in further generalizations of Aigner's result, and will be studying x^4+D y^4=z^4 for certain integers D over quadratic extensions of the rationals.

The student will perform the following tasks:

A. Study the following

Manley, E.D., On quadratic solutions of x^4=py^4=z^4, Rocky Mountain Journal of Mathematics, 36 #3 (2006) 1027--1031.

Mordell, L.J., The Diophantine equation x^4+y^4=1 in algebraic number fields, Acta Arith., 14 (1968/69) 347--355.

Mordell, L.J., The Diophantine equation x^2=y^4 +/- 1 in algebraic number fields, J. London Math. Soc., 44 (1969) 112--114.

Silverman and Tate, Rational Points on Elliptic Curves. UTM Springer 1992.

This will help the student become familiar with the question, provide some tools and understand how this question is connected to other questions regarding Diophantine equations.

B. Consider the various generalizations available and choose one. Solve the problem and write up a research report that will be the basis for a research paper to be submitted to a suitable journal.

C. Create a final research report to be presented at the MAM Symposium.

OTHER REQUIREMENTS: The students interested in the project above are expected to have mathematical maturity, a good understanding of basic number theory and algebra. They should become familiar LaTeX, (a popular markup text-editor for mathematical papers), as the project is under way.