Senior Projects 3: Eisenstein

Student #3 was another of those who took my Development of Modern Mathematics course this spring. I've had him in a number of classes, and he's pretty consistently been on top. He expressed an interest in algebra, so I decided to give him something challenging: the Quadratic Reciprocity Theorem.

I've mentioned this theorem before, as being one of the most beautiful in all of mathematics. It was conjectured by Leonhard Euler, but it was Karl Friedrich Gauss who first proved it. (Why, I wonder, does it seem OK to give his last name or all three, but "Karl Gauss" seems incomplete?) In fact, Gauss was so taken with the theorem that he came up with a number of different proofs of it. (And why not? Michelangelo sculpted four different Pietas, and each says something different about the subject; likewise Gauss's multiple proofs.) The proof most commonly given in number theory texts was discovered somewhat later by Ferdinand Eisenstein, who also devised analogous Cubic and Biquadratic Reciprocity theorems. (They're not as pretty as the QRT, for technical reasons, but they did point the way toward later, even broader extensions.) This student's task is to master and present Eisenstein's proof.

( Quadratic Reciprocity )