Department of Mathematical Sciences
Northern Illinois University

MATH 520, Fall 2007

12:00-12:50, M W F, DU 328

Professor John Beachy, Watson 355, 753-6753

Office Hours: 10:00-10:50 MWF in Watson 355, or by appointment

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About the midterm exam | About the final exam | Fall 04 Exams | Fall 98 Exams

Review Problems on Groups and Galois Theory, pdf format, 55 pages (review keyed to the undergrad text)


Group theory including the Sylow theorems, the basis theorem for finite Abelian groups. Polynomial rings, field theory, Galois theory, solvable groups, and solvability of equations by radicals.

PREREQUISITE: MATH 421 or consent of department.

TEXT: Lecture Notes; the textbook Algebra, by Hungerford, is recommended as a reference. It's published by Springer and they've kept the price down.

TENTATIVE SYLLABUS: I hope to cover almost everything in the class notes, with the possible exception of 2.8 (Nilpotent groups) and 4.6 (Computation of Galois groups). The primary material corresponds to Chapters 7 and 8 of Abstract Algebra (Beachy/Blair); some additional material and exercises have been added.

GRADING: Semester grades will be based on 500 points:

200 points for homework
100 points for a midterm exam
200 points for the final exam

FINAL EXAMINATION: The final exam is scheduled for Monday, December 10, 12:00-1:50


Week Day       Section  Topic
            | What actually happened
           \|/    Tentative schedule
 1   8/27  1.1    1.1   Definition, isomorphisms
     8/29  1.2    1.2   Subgroups, cyclic groups
     8/31  1.3    1.3   Semidirect products
 2   9/03            HOLIDAY
     8/05  1.3    1.3   Permutation groups
     9/07  1.4    1.4   Homomorphisms, factor groups, isomorphism theorems
 3   9/10  1.4    2.1   Automorphisms
     9/12  2.1    2.1   Semidirect products
     9/14         2.2   Conjugacy 
 4   9/17         2.2   Cauchy's theorem
     9/19         2.3   Group actions
     9/21         2.3   The class equation
 5   9/24         2.4   The Sylow theorems 
     9/26         2.4   The Sylow theorems 
     9/28            EXAM I
 6  10/01         2.5   Finite abelian groups
    10/03         2.5   Finite abelian groups
    10/05         2.6   Solvable groups
 7  10/08         2.6   The Jordan-Holder Theorem
    10/10         2.7   Simple groups
    10/12         2.7   Simple groups
 8  10/15         2.9   Groups of order < 16
    10/17         2.9   Groups of order < 16
    10/19            EXAM II
 9  10/22         3.1   Polynomials 
    10/24         3.2   Irreducible polynomials
    10/26         3.3   Algebraic extensions 
10  10/29         3.4   Splitting fields 
    10/31         3.4   Splitting fields 
    11/02         3.5   Finite fields 
11  11/05         3.5   Finite fields 
    11/07         4.1   Galois groups 
    11/09         4.1   Galois groups 
12  11/12         4.2   Separability
    11/14         4.2   Separability
    11/16         4.2   Separability 
13  11/19         4.3   The fundamental theorem 
    11/21            HOLIDAY
    11/23            HOLIDAY 
14  11/26         4.3   The fundamental theorem 
    11/28         4.3   The fundamental theorem 
    11/30         4.4   Solvability by radicals 
15  12/03         4.4   Solvability by radicals
    12/05         4.5   Cyclotomic extensions 
    12/07         4.5   Cyclotomic extensions 
16  12/10            FINAL EXAM   12:00-1:50 p.m.


Homework assignments::
  Hmwk 1   |   Hmwk 2   |   Hmwk 3 and Hmwk 4   |   Hmwk 6   |   Hmwk 9

Due date  Section  Problems

Dec 07          Homework 9

                Homework 8:
Nov 17    4.1      #1, 4, 5, 6
          4.2      #2, 7  

                Homework 7:
Nov 12    3.4      #3, 4, 7, 11, 12   (prove #7 for all n)
          3.5      #3, 5, 6, 12, 16

I encourage you to study in groups, and you may discuss homework problems with other students. You should write up your own solutions--direct copying is unacceptable. As a rough guideline for writing up solutions to homework problems, you should include enough detail so that (i) you can convince me that you understand the solution and (ii) you can understand your solution when you study for the final exam. Two assignments will be given under the rules for a takehome exam--no consultation with other students.


I wrote up notes when I taught 520-521 in 1992-93, and included quite a bit of preliminary material. You don't need to know Zorn's lemma this semester. The rest of the material should be familiar if you had Math 421 at NIU. Here is the table of contents:

     0.1 Sets, functions, and equivalence relations .....  5
     0.2 Permutations ................................... 12
     0.3 Construction of the number systems ............. 15
     0.4 The ring of integers ........................... 21
     0.5 The ring of integers modulo n .................. 24
     0.6 Vector spaces .................................. 29
     0.7 Matrices ....................................... 33
     0.8 Zorn's lemma ................................... 35

This material is available for you to use as a reference, in the form of a pdf file, located on the page with the Lecture Notes.


Hungerford, Algebra
This is an excellent reference book, and has been a standard since it was originally published in 1974. The down side is that students seem to find it difficult to learn material here for the first time. Many of the proofs say "sketch of proof" and you have to work to figure out what was left out.

Dummit and Foote, Abstract Algebra
This book has a wealth of examples, and so it is an excellent reference book. In 1998 I used it for the first half of 520, and then found it did not fit our syllabus for the second half. My one criticism is that so much material is included that it seems to be hard for students to decide what is really important. I would also like the first part to be written a little more tightly, like other graduate textbooks. It is quite a bit more expensive than Hungerford.

Clark, Elements of Abstract Algebra
This is a Dover paperback, so it is a great value. I recommend it for a slightly different point of view, while being at about the level of our class notes.

Jacobson, Basic Algebra I, II
An excellent book by a very famous ring theorist. It is my personal choice for a textbook, but it contains more than we need for 520 and 521, and would mean buying two books (I wish I could write half as well as Jacobson could.)

Rotman, Advanced Modern Algebra
This is a new book, but Rotman is an experienced author, and has a good choice of material. Be careful--that last time it was used here the students found lots of mistakes.

van der Waerden, Algebra, volumes I and II
Based in part on lectures in the 1920's by two giants in the field, Emmy Noether and Emil Artin, it shows its age even though it has been updated. It is still an important reference, and contains much that is of historical interest.

Lang, Algebra
Generally judged to be more encyclopedic and harder than Hungerford. Many schools use it as a reference, while the professor depends mostly on notes in class.

Rotman, Introduction to the theory of groups
I'm familiar with a much earlier version, but I believe this should still be a good reference.

Garling, A Course in Galois Theory
Paperback; gives a good introduction to Galois theory.

Herstein, Topics in Algebra
A classic, ostensibly written for undergraduates. Herstein's distinctive point of view makes it interesting reading; he was a master teacher. The problem sets are superb (some are very difficult).


Factoring polynomials | Abstract Algebra OnLine | Group tables | Group tables 2 | General information about 520

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