Test on Chapters 4-6

Be able to do:

• Polynomial arithmetic on polynomials with coefficients in a given ring.
• Long divide polynomials with coefficients in a given field.
• Use long division to describe congruence classes in a polynomial ring mod a given polynomial.
• Show that Q(√ n ) is a field (chapter 5)
• ((Use the first isomorphism theorem on the function f : Q[x]→ Q(√ n ) given by f(p(x))=p(√ n ) to prove that Q[x]/(x²-n) is isomorphic to Q(√ n )))--deleted
• prove that a subset of a ring is (or is not) an ideal
• List the principal ideals of a ring
• List the cosets of an ideal in a ring
• prove that a function is a surjective homomorphism
• find the kernel of a homomorphism.
• Prove a map from Z_nm → Z_n is a well defined homomorphism
• Use the first isomorphism theorem to prove an isomorphism
• Use thm 6.11 to prove a homomorphism is 1-1.

Know how to prove:

Theorems 6.2, 6.4 and 6.5