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:



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






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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