Math 361, Spring 2013, Assignment 12

Carefully define the following terms, then give one example and one non-example of each:

1. Unit.
2. Associate.
3. Irreducible element.
4. Prime element.
5. Unique factorization domain.
6. Principal ideal domain.
7. Euclidean norm.
8. Euclidean domain.

Carefully state the following theorems (you need not prove them):

1. Theorem characterizing unique factorization domains (given in class; this does not appear explicitly in the text).
2. Theorem concerning polynomials over a UFD (Theorem 45.29 in the text).
3. Theorem on unique factorization in PIDs (Theorem 45.17 in the text).
4. Theorem on ideals in Euclidean domains (Theorem 46.4 in the text).

Do the following problems:

• Section 45, problems 1, 3, 5, 7, and 9.
• Section 46, problems 1, 3, and 5.