Math 3500 Modern Algebra I
Spring 2011
Final Exam Material
Chapters: 0-7, 9-10, 12, 14 Parts of: 8, 13, 16
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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
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Groups
- definitions and basic properties: order, identity, inverses, etc.
- cyclic, symmetric, dihedral, and abelian groups, Z/nZ, U(n), Z, GL2(R), SL2(R)
- subgroups
- center and centralizer
- direct products
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Mappings
- isomorphisms and homomorphisms
- automorphisms
- cosets and normal subgroups
- Lagrange's Theorem and consequences
- factor groups
- 1st Isomorphism Theorem and consequences
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Rings
- definitions and basic properties
- Z/nZ, Z, M2(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