[FOM] Dirichlet's theorem; boiling down proofs

Wayne Aitken waitken at csusm.edu
Wed Aug 22 15:35:39 EDT 2007

Jacques Carette wrote:

> My undergraduate degree (in pure mathematics, completed a mere 17 years
> ago) covered the last 5 topics in decent depth, and half of the first
> topic, but no projective geometry.  And in fact, the proofs were based
> on exactly the tools you mention

Also a talented, ambitious undergraduate in a good program can and should
take graduate courses, read on his/her own, and learn informally from
faculty and graduate students. So I will agree that all 19th century
mathematics, and a large amount of 20th century mathematics is within
reach of such an undergraduate.

Furthermore, as Jacques Carette mentions, undergraduates can (and should)
get a decent introduction to several areas of 19th century mathematics.
For example, introductory galois theory is a staple of undergraduate
algebra, Körner's Fourier analysis is written for undergraduates, and
several undergraduate level geometry texts introduce projective geometry.

Realistically, however, outside of core areas such as analysis and
algebra, an undergraduate curriculum has room for only a one semester
course in any given area. My point is that there are 19th century results
whose proof would be inappropriate for such a course. For example, it
would not be a good idea to give a full proof of the Kronecker-Weber
theorem in an introductory one semester undergraduate course in algebraic
number theory. The results of the Italian school of algebraic geometry
give even stronger examples. Given the large amount of 19th century
research in projective geometry, invariant theory, ellipic and abelian
integrals, and Fourier analysis, I suspect that there are results in these
areas whose proofs would similarly be out of place in a one semester
introductory undergraduate course even at a top 25 program.

Now, Dirichlet's theorem is different. Its proof could be covered in an
introductory analytic number theory course. After all, it is the start of
an extremely fruitful tradition (culminating in the Langland's program).

    --- Wayne Aitken

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