Most of my days seem to be rather dull. I teach twice a week, and spend most of my time sitting in my office working at my laptop. Rarely do I have any visitors, and rarely do I leave to talk to others. I know some would call this a peaceful life. I prefer to call it a solitary existence. But today was different.

I thought it would be a quiet day because I was going to hold office hours to help review for my midterm tomorrow. I figured no one would show up because my differential equations course has been pretty simple so far. In fact, I got just a handful of students who asked just a handful of questions.

You see, then Radhika Ganapathy stopped by to talk about some topics in our reading course on Algebraic Number Theory. She asked a really simple question, and I feel like an idiot because I couldn't give her an answer right away:

"Let A be Dedekind ring which has unique maximal ideal P. Then P is a principal ideal."

The idea of the proof is simple: Pick an element x in P which is not in P^2; it suffices to show P = (x). Clearly (x) < P, so it suffices to show that any y in P is in the form y = x z for some z in A. I'm probably going to have to resort to reading through Atiyah and MacDonald. Why couldn't I remember the proof?!? Maybe my mind was toast after talking about the Method of Undetermined Coefficients.

Then after lunch, Navin Dasigi stopped by to ask about the Riemann Hypothesis for elliptic curves over finite fields. I told him it wasn't really all that interesting because Deligne's proof of the Weil conjectures tell you explicitly what all of the zeroes are, but then I realized the political issue of Louis de Branges' work the Riemann Hypothesis for the classical Riemann zeta function... I was quick to discuss the heurestic of saying how the zeroes of the local zeta function should give information about the global zeta function, so the global case is still very interesting.

Then after he left, Alain Togbe stopped by to talk about our joint paper on solutions to a certain family of sextic Thue equations. I've been thinking about some of the questions that have been floating around on the Number Theory ListServ recently, so I thought I'd pose the following question to him:

"Noether's Problem asks when given a finite group G and the function field K = Q(t), there exists an normal extension L = Q(x) of K such that Gal(L/K) = G. When G = Z_3 is cyclic of order 3, is it necessarily true that L = K(x) is defined by the polynomial p(x) = x^3 - t*x^2 - (t+3)*x-1?"

I believe the answer is yes, but I need to think about this tonight. I can probably work this out without too much trouble.

Then after Alain left, Joe Lipman stopped by to talk about some of the topics he's covering in his course on modular forms. We were wondering about the "universal elliptic curve" which is the modular curve X(N) which parametrizes triples (E,P,Q) of curves E where {P,Q} is a basis for the collection E[N] of N-torsion. We agreed that the function field Q( j(z) ) is easy to work with for X(1), but that X(N) is a bit more complicated.

Whew! What a day. My mind is completely wasted. I just came from sitting in Pappy's enjoying a triple cheeseburger, trying not to think about anything at all. Except that solution I worked out in my sleep last night for this Diophantine equation...