USNA Mathematics Department seminar

Title:

Abstract: In the late 1630s, Pierre Fermat wrote that the equation xⁿ + yⁿ = zⁿ is not satisfied by natural numbers if n>2. The proof of this statement was given convincingly by Andrew Wiles in 1995. He used the latest methods from arithmetic geometry involving elliptic curves and modular forms. But before Frey showed how to apply these techniques in 1985, most of the progress on solving Fermat's last theorem was made using algebraic number theory, specifically cyclotomic field theory. This theory more or less originated in the work of Kummer and was used by him and others to solve many special cases of Fermat's last theorem. My talk will start with a brief discussion of Wiles' approach and then focus on the approach of Kummer and others using cyclotomic field theory, and end with a brief discussion of the current status of this theory.