Math 818: Problem set 6, due February 25

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, February 25, 2005. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
  1. Let R be the reals. Show that the polynomial ring R[x,y] in two variables is not a PID.
  2. Let R be a domain, and let p be a nonzero nonunit element of R.
  3. Let A be a finite abelian group. Let |A| = mn, where m and n have no common factors. Let Am = {a in A | ma = 0}, and let An = {a in A | na = 0}.
  4. Let R = F[x], where F is a field. Show that there are infinitely many irreducible elements in R. [Hint: Euclid's proof that there are infinitely many primes in the integers works here!]