[FOM] Dirichlet's theorem; boiling down proofs

Timothy Y. Chow tchow at alum.mit.edu
Mon Aug 20 17:29:38 EDT 2007

Joe Shipman wrote:
>Yes, unless there is another counterexample (Dirichlet's theorem is the 
>only unarguable counterexample so far, though I am currently researching 
>this to see how much of it can be made accessible).

I'm curious as to how you came to the conclusion that Dirichlet's theorem 
is an "unarguable" counterexample.  What exactly is your definition of 
"undergraduate"?  When I was an undergraduate at Princeton, there was a 
junior seminar that went through about 2/3 of Serre's "Course in 
Arithmetic," including the proof of Dirchlet's theorem.  Another junior 
seminar went through the classification of finite-dimensional simple Lie 
algebras over C.  A senior-level class taught by Shimura covered a 
significant amount of algebraic and analytic number theory, including the 
statements (but not proofs) of the main theorems of class field theory.

Princeton isn't typical, of course, and I wouldn't call class field theory 
an undergraduate subject just because there exists one undergraduate 
course somewhere that covers it.  But if you're going to use words like 
"unarguable" then I would like to understand your ground rules better.

I'll also repeat that the question of rigor is important.  Riemann-Roch 
for surfaces is surely beyond the reach of today's undergraduates.  Does 
it fail to count because the 19th century version wasn't rigorous enough 
by modern standards?

By the way, if you meant 100 years as a general rule, then I have to admit 
that I'm not expecting SGA4 to be accessible to undergraduates by 2072.  
Unfortunately few of us are likely to still be around to check.


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