Mathematics 4321                       Review Test #2                             Spring 2007

Dr. Cordero

1. Define the ring of polynomials over a ring R.

2. Give two properties of R which are inherited by R[x].

3. Under what conditions is R[x] a field?

4. State the Division Algorithm for rings of polynomials. Give an example showing how to use the division algorithm.

5. State and prove the Remainder Theorem. Give an example that uses the Remainder Theorem.

6. What does it mean f(x) | g(x)?

7. State and prove the Factor Theorem. Give an example that uses the Factor Theorem.

8. Give the definition of the greatest common divisor of two polynomials f(x) and g(x) over a ring R. How can the gcd of two polynomials f(x) and g(x) be found? How can the gcd of two polynomials f(x) and g(x)  be written as a linear combination of f(x) and g(x)?

9. Define irreducible polynomial; give examples in Q[x], R[x] and .

10. State the Unique Factorization Theorem.

11. Define ring homomorphism and give an example.

12. Define kernel of a ring homomorphism.

13. Let  be a ring homomorphism.

• Show  is a subring of S.
• Show  is one-to-one iff Ker={}.

14. Do all assigned homework problems from sections 32, 34, 35, 36, 38.