[FOM] Dirichlet's theorem; boiling down proofs

Wayne Aitken waitken at csusm.edu
Sun Aug 19 17:57:54 EDT 2007

> I propose the thesis "any mathematics result more than a century old is
> suitable for undergraduate math majors".

> So far I have only found one significant counterexample, Dirichlet's
> theorem (which, in its logically simplest form, states that if a is
> prime to b, there exists a prime congruent to a mod b).

One could design an accessible senior seminar on analytic number theory
that covered Dirichlet's theorem, the prime number theorem, and several
other classic topics. The prerequisites would include standard
undergraduate courses in complex analysis and number theory.

On the other hand, I suspect one could find counter-examples to your
thesis in the following areas, all already highly developed in the 19th

* Elliptic and abelian integrals, and the associated theory of elliptic
curves and jacobian varieties.
* Projective geometry.
* The differential geometry of Gauss and Riemann.
* Galois theory.
* Algebraic number theory.
* Algebraic geometry (especially the Italian school).
* Fourier analysis.

In many cases the original 19th century proofs were complex, unnatural,
and sometimes lacking in rigor, and the best modern proofs use the
(largely graduate level) mathematical machinery of the 20th century:
measure theory, functional analysis, point set topology, more abstract
forms of linear algebra, commutative algebra, homological algebra, and so
on. In fact, one impetus for the development of the 20th century machinery
was to provide more elegant foundations for the 19th century results.

   --- Wayne Aitken

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