Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:     Do get started on your Reflections on the History of Mathematics paper.
  You will note that the rolling break due to spring breaks continues; the exercises should be a little lighter again this week.
      The theme this week is the the beginning of modern algebra and the start of advanced number theory. It starts with the revival of elementary number theory, particularly in the work of Fermat. There is a decent discussion of his little and last Theorems. Both of these are elementary, and his 'little theorem' underlies one of the great technological advances in the 20th century—the development of public-key cryptography. The last Section 11.6 introduces us to a completely new phenomenon—curves in the plane that cannot be parametrized by rational functions. This leads to the problem of how you represent them explicitly (as the set of zeroes of some equation is an implicit representation. It turns out that curves defined by cubics and by quartics of the form y²=f(x), where f has degree 4, admit a geometric method to construct points on the curve that turns the curve into an abelian group, but that is a later topic.
  We next read the story of the start of more advanced number theory in the theory of elliptic integrals and their transformations–this is for background when we discuss complex elliptic curves. Elliptic integrals are quite important historically; they led to some of the most intense mathematics of the 19th century. Initially they are hard both conceptually (they are integral along real paths in the complex plane), as well as technically, as the transformation rules involve manipulating expressions in the limits of the integrals.
      It is worth noting that in the first half of the 19th century the study of elliptic integrals and elliptic functions was one of the main topics in mathematics. While terminology from that time persists, this is no longer an area of current research, although it informs much of modern mathematics, from algebraic geometry to the proof of Fermat's last Theorem (Wiles' Theorem) to the extraordinarily deep Langlands program that lies at the interface of number theory, analysis, and representation theory.
      The story of Fermat's Last Theorem was big news some 30+ years ago. When Andrew Wiles announced his proof (at a conference in Cambridge) and released a manuscript, mathematicians all over the world started to study it for Wiles was a widely-known and highly respected mathematician—and not a crank (There is a continuous rain of papers purporting to prove famous results with elementary means, almost all of which have easily detected flaws; These are typically sent to famous mathematicians. When I was a student at Cambridge there was a bulletin board in the mathematics department common room devoted to letters from mathematical cranks, and I have even received several such manuscripts and letters.) After Wiles' announcement, some faculty at the University of Chicago set up a study group, where we read his paper and lectured on it and the background material for most of a year; this fell apart when Wiles retracted his paper, having found an error that he later corrected with the help of Richard Taylor.
• Reading:
  • Chapter 11 in Stillwell's book. This covers the revival of number theory and some great little gems, such as Fermat's little theorem (which is the basis for one of the first public-key cryptosystems), as well as his famous Last Theorem (Section 11.3).
  • Chapter 12 in Stillwell's book. Some of the topics in this book are more advanced (e.g. the transformations of elliptic integrals and how it leads to periodic functions with two independent periods), so I will not assign any exercises.
• Assignment: Due Monday, 11 March 2024. (HW 8) Here is a .pdf and a LaTeX source of the assignment.
  To hand in: We are using Gradescope for homework submission.
  1. Exercises 11.2.1, 11.2.2, and 11.2.3 in Stillwell. These give a refreshingly elementary proof of Fermat's Little theorem.
  2. Exercise 11.4.4. For this, recall the parametrization of Pythagorean triples. Use that to deduce Fermat's last Theorem in the case of n=4.
  3. Exercise 11.6.1, 11.6.2, and 11.6.3. These together, with Exercise 11.4.4 prove the statement about the impossibility of a rational parameterization of y²=1-x⁴. This works because the integers, like polynomials in one variable with coefficients from a field, form a Principal Ideal Domain, and such rings have unique factorization in to prime/irreducible elements.

Last modified: Mon Mar 11 11:17:02 PDT 2024 by sottile