In the first course (Algebraic structures, 89-214), you were supposed to cover the following:
* Introduction to number theory: division, Euclid algorithm, gcd.
* Definition of group. subgroups. Euler group. Cyclic groups.
* Symmetric groups and dihedral groups. Cosets and Lagrange theorem.
* Normal subgroups. Quotient groups.
* Isomorphism theorems.
* Rings: definitions, division rings, fields, ideals, principal ideals, integral domain, prime and maximal ideals.
* Quotient ring and isomorphism theorems. Chinese remainder theorem.
We covered the following topics:
* Group actions
* Sylow subgroups
* p-groups
* Euclidean domains, unique factorization
* Galois theory, including the main theorem and solvability by radicals (ignoring separability issues and assuming roots of unity)
* Finite fields
You may have a look at my exercise booklet on groups (86 pp.), and my exercise booklet on rings (50 pp.). Both are more suitable for a full semester length course on the respecitve topics, but can be used here as well. Note in particular chapter 12 of the first booklet, and chapter 5 in the second.
|
