Harbourne's Math 818 Problem set 6

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.)

Let R be the reals. Show that the polynomial ring R[x,y] in two variables is not a PID.
- (a) Find a prime factorization of 42 + 56i in the Gaussian integers. Indicate briefly how you did it.
- (b) Write 42+56i as (4+i)(a+bi) + (r+si), where a, b, r and s are integers, and either r+si is 0 or |r+si|² < |4+i|². Indicate briefly how you did it.
Let R be a domain, and let p be a nonzero nonunit element of R.
- (a) Show p is irreducible if and only if the only principal ideal properly containing (p) is R itself.
- (b) Show p is prime if and only if (p) is a prime ideal.
Let A be a finite abelian group. Let |A| = mn, where m and n have no common factors. Let A_m = {a in A | ma = 0}, and let A_n = {a in A | na = 0}.
- (a) Show that A is isomorphic to A_m x A_n. (This is essentially the Chinese Remainder Theorem, again!)
- (b) Show that if r is the maximum order among elements of A, then ra = 0 for all elements a of A. [Hint: use induction on |A|.]
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!]