＊Contents (I, II)：
In this course we want to study the basic theory of algebra, containing groups, rings and some basic
field theory with applications.
(A) Basic Group Theory
1. Definition of a Group
2. Examples of Groups
3.Subgroups, Lagrange's Theorem
4. A Counting Principle
5. Normal Subgroups and Quotient Groups
6. Homomorphisms, Cauchy's Theorem
7. Automorphisms
8. Cayley’s Theorem
9. Permutation Groups, Odd and Even Permutations, Cycle Decomposition
10. Another Counting Principle
11. Conjugacy and Sylow's Theorem
12. Finite Abelian Groups
(B) Basic Ring Theory
1. Definitions and Examples
2. Ideals, Homomorphisms, and Quotient Rings
3. The field of Quotients of an Integral Domain
4. PIDs, Euclidean Rings, UFDs
5. Polynomial Rings
(C) Fields
1. Examples of Fields
2. Field Extensions
3. Finite Extensions 4. Constructions with Straightedge and Compass
5. The Elements of Galois Theory
6. Solvability by Radicals
7. Galois Groups over the Rationals