Analogies between integers and polynomials

In late July, 2013, I had the of privilege of delivering three two-hour lectures at the University of the Philippines Diliman, as part of a CIMPA-ICTP research school. The lectures discussed the analogies between the ring of integers Z and the ring F_q[T] of polynomials over a finite field. Towards the second half of the course, we focused on analogies in prime number theory. All that was assumed was a working knowledge of the theory of finite fields.

For students of the program, and all other interested parties, I include my lecture notes here.

• Lecture notes for hours 1--3 [PDF]
  Topics include quadratic reciprocity for polynomials (in odd characteristic), Fermat's last theorem for polynomials and Mason's theorem, and the resolution by R.E.A.C. Paley of an analogue of Waring's problem.
• Lecture notes for hours 4--6 [PDF]
  We started by discussing analogies between the classical prime number theorem and Gauss's formula for the number of irreducible polynomials of a given degree. We then proved Carlitz's theorem counting self-reciprocal (palindromic) irreducibles. The last part of the course focused on polynomial versions of the twin prime conjecture and the infinitude of primes of the form n²+1.
• Bibliography [PDF]