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-Hölder 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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