MATHEMATICS W4042, Fall 2015

Introduction to Modern Algebra II



This is the second part of the Modern Algebra sequence.  The main topics are rings, especially polynomial rings, and Galois theory.  




Provisional syllabus:  Each of the topics listed below will occupy roughly one-two weeks of course time. 


1.  Rings and ideals, basic notions

2.  Polynomial rings

3.  Modules (basic notions)

4.  Fields of fractions

5.  Principal ideal domains, polynomials over a field

6.  Irreducible polynomials and factorization, Eisenstein polynomials

7.  Field extensions and splitting fields

8.  Galois groups and the main theorems of Galois theory

9.  Applications:  finite fields, cyclotomic fields

10.  Applications:  solution by radicals, ruler and compass constructions


If time permits, we will cover noetherian rings and modules over a PID.



Prerequisites:  Modern Algebra I.


Textbook:  Joseph Rotman, Galois Theory.


The book Abstract Algebra by Dummit and Foote (on reserve in the math library) can be used as a reference. 


Online resources:


Abstract Algebra: Theory and Applications, by Thomas W. Judson


Notes on Modern Algebra II by Patrick Gallagher


Lecture notes by Robert Friedman


Modern Algebra II review notes by Robert Friedman in three parts: Part 1   Part 2   Part 3



Midterms:  October 13,  November 12 (in class)

Final:  to be announced

Practice exams and notes in various languages (as requested)


            Homework assignments

1st week  (due September 17)      

2nd week (due September 24)      

3rd week (due October 1)            

4th week (due October 8)

5th week (due October 15)

6th week (due October 22)

7th week (due October 29)

8th week (due November 5)

             (Second Midterm:  no homework)

9th week (due November 19)

10th week (due December 3)

11th week (due December 10)


            Solution sheets

1st week           

2nd week           

3rd week

4th week

5th week

6th week           

7th week

8th week

9th week

10th week

11th week


Notes on the Main Theorem of Galois Theory


Notes (of Z. Dancso and S. Morgan) on Ruler-Compass constructions vs. Origami constructions






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