SYLLABUS MATH 690Z

Galois Theory and Commutative Algebra

• Prerequisites: MATH504 and MATH 505

• Textbook: Hungerford,T. Algebra Springer-Verlag, New York-Berlin, 1980

• Objective: Understand the basic concepts and theorems in Galois Theory and commutative algebra. Being able to use the basic techniques to solve problems and prove theorems.

• Tentative Course Schedule:

	Topics	Time period (in class)
Chapter 1	Fields and Galois Theory	5 (weeks)
	Field extensions
	Fundamental Theorem
	Splitting fields
	The Galois group of a polynomial
	Finite fields
	Separability
	Cyclic, cyclotomic, radical extensions
Chapter 2	Commutative rings and modules	5 (weeks)
	Chain conditions
	Prime and primary ideals
	Primary decomposition
	Noetherian rings and modules
	Ring extensions
	Dedekind domains
	Hilebr Nullstellensatz
Chapter 3	Algebraic number fields	3 (weeks)
	Algebraic number fields
	Unique factorization in algebraic number fields
	Ramification and degree

• Grading: The grade is mainly based on homework assignments. Students are encouraged to study and work together. Most of the assignments will be discussed in class. Some projects will be suggested for interested students.

• Reference:

1. Jacobson, Nathan Basic algebra. I. W. H. Freeman and Co., San Francisco, Calif., 1974

2. Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969

3. Lang, S. Algebra Texts in Mathematics 211 Springer-Verlag, New York, 2002

4. Zariski, Oscar; Samuel, Pierre Commutative algebra. Vol. 1 Graduate Texts in Mathematics, No. 28. Springer-Verlag, New York-Heidelberg-Berlin, 1975

5. Zariski, Oscar; Samuel, Pierre Commutative algebra. Vol. II. Graduate Texts in Mathematics, Vol. 29. Springer-Verlag, New York-Heidelberg, 1975

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