MATH 124

Final Exam Topics

The final exam will be 8:00-10:00AM on Thursday, December 12, 2002.
The exam will cover everything discussed during the semester,
but will focus more on topics since the second exam.  Thus, you
should review the earlier material (Section 0.2, Chapter 1,
Sections 2.1, 2.2, and 2.3) as well as the following.
Since the second exam, we've studied sections 3.1, 3.2, 5.1, 5.2,
pieces of 5.5 and 5.6 (def 5.5.1, thm 5.5.2, thm 5.6.1, cor 5.6.2,
def 5.6.7, thm 5.6.10), 6.1, and 6.2.  Here is a list of the
major topics since the second exam:

Group homomorphisms
Kernel of a group homomorphism
Normal subgroup
   Know how to prove a subgroup is normal.
Factor group
Rings
Ring homomorphisms
Ring isomorphism
Types of rings
   Commutative
   Units
   Division ring
   Fields
      Z_n is a field iff n is prime.
   Zero divisors
   Integral domain
Polynomials
   Testing if a degree 2 or 3 polynomial is
     irreducible over a finite field.
Kernel of a ring homomorphism
Ideal
   Know how to prove a subset of a ring is an ideal.
Factor ring (also called quotient ring)
Prime ideals and maximal ideals
   Know what type of factor rings they produce.
Ideals in F[x] (polynomials over a field)
   Principal ideal generated by an element.
   Every ideal in F[x] is principal.
   A principal ideal in F[x] is maximal iff
     the generating polynomial is irreducible.
Finite Fields
   You can create a finite field as a factor ring
   of Z_p[x] where you use a principal ideal generated
   by an irreducible polynomial.
   
Also, looking over the homework problems can remind you
of the type of definitions and proofs you're expected
to know.