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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