**MTH 601: Algebra I (4)**

Monoids, Groups, group actions, Sylow's theorems, Finitely generated abelian groups, free groups

Rings and homomorphisms, Chinese remainder theorem, examples as polynomial ring and power series ring, rings of endomorphisms, Universal property of polynomial rings, Localization, Principal and factorial rings

Modules, quotient modules, direct product and direct sum of modules, Jordan-HÃ¶lder theorem, Free, Projective and Injective modules, Dual modules, Modules over PID

Category and Functor, Direct and Inverse limit

Polynomials in one and several variables, Gauss lemma, Irreducibility criterions, Power series ring, group of units in power series ring

Algebraic extensions, Algebraic closure, Splitting fields, Normal extensions, Separable and inseparable extensions, Finite fields

Galois extensions, examples, Galois correspondence theorems, solvability of equations

*Suggested Books*:

- Algebra (3rd Edition), Serge Lang, Addison Welsey
- Basic Algebra, Jacobson, parts-I and II, Dover Publications Inc.; 2nd edition.
- Algebra, Birkhoff and McLane, Chelsea Publishing Co.
- A course in theory of groups, D.J.S. Robinson, Springer; 2nd ed.

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