Friday, January 27, 2006

Overview of the background to Kummer's Proof

Ernst Kummer's proof for Fermat's Last Theorem was a major breakthrough in the history of number theory.

To appreciate the importance of Kummer's work with complex numbers, it is valuable to review earlier developments. Kummer came to his discoveries by trying to generalize quadratic reciprocity so it makes sense to review the details behind the basic proof for quadratic reciprocity. Lamé's work used cyclotomic integers which are derived from the roots of unity. For this reason, I will review basic proofs about π and Euler's famous equation: e = -1.

Kummer's work became the foundation for algebraic number theory. From this view, it makes sense to follow up Kummer's proof with the work of Abel, Galois, as well as the transcendental nature of π.

With this background, it makes sense to go through Dedekind's reinterpretation of Kummer's work which became the foundation of modern algebraic number theory.

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