Spring 2007 – Mathematics 4321 – Review Test #2

— 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={}.
  1. Define ideals and principal ideals and give an example.
  2. Study the statement and  proof of Theorem 38.1.
  3. Define  quotient ring.
  4. Show that every ideal is a kernel. (Theorem 39.2)
  5. State and prove the Fundamental Homomorphism Theorem for Rings.
  6. Define prime ideals and maximal ideals and give an example.
  7. Study Theorems  39.8 and 39.12.
  8. Do all assigned homework problems from sections 34, 35, 36, 38, and 39.