### Math 818: Problem set 7,
due March 11

*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, March 11, 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.)
- A polynomial ring R[x] is a UFD if R is, by
Theorem 3.9, p. 401. Show by example that R[x] need not be
a Euclidean domain, even if R is. Justify your example:
how do you know your R is a Euclidean domain and
how do you know R[x] is not?
- For any subring R of the complex numbers, f(r) = rr*
(where r* is complex conjugation) defines a size function on R.
Suppose R =
**Z**[d^{1/2}i], where d is an integer. Prove that
f makes R into a Euclidean domain for d = 1 and d = 2,
and show where the proof fails if d > 2.
- Prove that two integer polynomials f and g are relatively prime in
**Q**[x] if and only if the ideal they generate in
**Z**[x] contains a positive integer.
- Prove that the kernel of the homomorphism
**Z**[x] -> **R**
sending x to the real number 1 + (2^{1/2}) is principal,
and find a generator for this ideal.
- Show each of the following polynomials is irreducible in
**Q**[x].
- (a) x
^{6} - 2x^{3} + 3x^{2} - 1
- (b) x
^{3} + 5x^{2} + 10x + 5
- (c) x
^{2} + x + 2
- (d) x
^{2} + 2345x + 125