Math 3500 Modern Algebra I
Spring 2011
Final Exam Material
Chapters: 07, 910, 12, 14 Parts of: 8, 13, 16

Background Material
 equivalence classes
 functions/mappings
 properties of the integers (division algorithm, least common multiple, greatest common divisor, etc.)
 modular arithmetic
 proof by contradiction, if and only if proofs, contrapositive, "or" proofs, induction

Groups
 definitions and basic properties: order, identity, inverses, etc.
 cyclic, symmetric, dihedral, and abelian groups, Z/nZ, U(n), Z, GL_{2}(R), SL_{2}(R)
 subgroups
 center and centralizer
 direct products

Mappings
 isomorphisms and homomorphisms
 automorphisms
 cosets and normal subgroups
 Lagrange's Theorem and consequences
 factor groups
 1st Isomorphism Theorem and consequences

Rings
 definitions and basic properties
 Z/nZ, Z, M_{2}(R), Z[i], subrings
 zero divisors and integral domains
 fields: Z/pZ, R, C, Q
 ideals, prime ideals, maximal ideals
 factor rings, examples with polynomial rings
 polynomial rings, Theorem 16.1, division algorithm
Last Updated: April 26, 2011