MATH 3163 Course Outline

MATH 3163 Modern Algebra Course Outline

Week 1: an overview of ring theory; a proof of the division algorithm for integers Week 2: divisibility, prime numbers, and factorization; the fundamental theorem of arithmetic Week 3: congruence and congruence classes; modular arithmetic, and a construction of the rings Z_n Week 4: more on modular arithmetic, with a particular emphasis on the difference between Z_p (when p is prime) and Z_n (when n is composite) Week 5: review; first exam Week 6: abstract ring theory; examples of different kinds of rings; basic properties of integral domains and fields Week 7: subrings and product rings; isomorphisms and homomorphisms Week 8: polynomial rings of the form R[x]; the division algorithm in F[x] when F is a field Week 9: divisibility, reducibility, and factorization in F[x]; the greatest common divisor theorem for polynomials Week 10: review; second exam Week 11: more on irreducible polynomials; a proof that every polynomial in F[x] factors in an essentially unique way into irreducibles Week 12: congruence classes in F[x], and a definition of the ring F[x]/(p(x)) Week 13: properties of the ring F[x]/(p(x)), including a proof that it is a ring extending F Week 14: the structure of F[x]/(p(x)) when p(x) is irreducible Week 15: review; third exam Week 16: review for final exam; final exam