### MA 5320 Algebra I

#### Course Details

Group Theory: Review of basic Group Theory, Group Actions, Kernel and Stabilizer of Group Actions, Transitive Group Action, Cayley's Theorem, The Class Equation, Sylow's Theorems, Direct Products, Structure Theorem for Finite Abelian Groups, Existence and universal Properties of free Groups, Examples of Groups specified by Generators and Relations.

Ring Theory: Review of basic Ring Theory, Properties of Ideals, Prime and Maximal Ideals, Two-sided ideals and Quotient Rings, Chinese Reminder Theorem, Euclidean Domain, Euclidean Algorithm, Principal Ideal Domain, Euclidean Domain is a Principal Ideal Domain, UFD, PID implies UFD, Universal Property of a Polynomial Ring, Criteria for Irreducibility. Definition and simple examples of modules over commutative and non-commutative rings.

Field Theory: Finite and Algebraic Extensions, Existence and Cardinality of Algebraic Closure, Finite Fields, Galois Theory of Polynomial in characteristic zero and simple examples.

#### Course References:

Text Books:

1. D. S. Dummit and R. M. Foote: Abstract Algebra, 2nd Edition, John-Wiley, 1999.

2. S. Lang: Algebra 3rd Edition, Addison-Wesley, 1999.

Reference:

1. J.A. Gallian: Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.

2. M. Artin: Algebra, Prentice Hall inc 1994.

3. I.N. Herstein: Topics in Algebra, John-Wiley, 1995.

4. T. A. Hungerford: Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, 1980.