MATH 430 : Modern Algebra (Winter 2014):


Instructor: Alvaro Pelayo, apelayo@math.wustl.edu

Required Text: John B. Fraleigh, "A First Course in Abstract Algebra, 7th Edition", Addison-Wesley, 2003.

Other texts:

1. Joseph J. Rotman, "A First Course in Abstract Algebra, 2nd Edition", Prentice Hall, 2000.

2. Michel Artin, "Algebra", Prentice Hall, 1991.

3. I.N. Herstein, "Abstract Algebra, 3rd edition", Prentice Hall, 1996.

Grading: homework (15%), Midterm Exam 1 on Wednesday February 19, 2014 (25%), Midterm Exam 2 on Monday March 24, 2014 (25%), cumulative Final Exam (35%) on May 7, 2014. Exams will consist of a few theory questions, including definitions and proofs of selected results, and some problems involving computations and proofs. There will be no make-up exams - if you miss one midterm, then final exam counts 60%. If you take both midterms and your grade in the final is greater than your lowest midterm grade, then the grade in the midterm gets replaced by the grade in the final. For example, if midterm 1 score is 80/100, midterm 2 score is 50/100 and final score is 70/100, your score in the second midterm gets replaced by 70/100. If you choose to be graded "Pass/Fail", a "Pass" grade reqires a grade of C- or higher.

Homework: Assignments can be downloaded from this website, no paper copy will be given in class. A grader will grade selected problems. Discussing homework with others is ok. It is expected that everyone writes in his/her own words the homework solutions. No late homework is accepted and the two lowest homework grades will be dropped (which can also count for missing assignments). Homework is due at the beginning of class on the due date.

Syllabus: Sets and relations. The integers, congruences and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions - as in (parts of) chapters 0-9 of the book.

Outline: about 6 1/2 weeks spent on "preliminaries" and "group theory" (section 0 and chapters 1,2,3,7), and 6 1/2 weeks on "ring theory and fields" (chapters 4,5,6,9). There will be no time to cover all of the sections in each chapter and not all sections will be covered in the same depth. Class time is for the fundamental concepts of each chapter. Fraleigh's book (and also the other books) are sources for further explanations and examples. The following is the plan *subject to changes* (numbers refer to sections in Fraleigh's book); this plan will be updated as the course progresses.


WEEK 1:

Lecture 1 (Mo January 13): 0, 1 (Sets, maps, cardinality, equivalence relations) Homework 1 (sec. 0,2,3,4)

Lecture 2 (We January 15): 2 (Binary operations)

Lecture 3 (Fr January 17): 3 (Isomorphism between binary operations)


WEEK 2:

Monday January 20 is a holiday

Lecture 4 (We January 22): 4 (Groups)

Lecture 5 (Fr January 24): 4 (Groups, continuation)


WEEK 3:

Lecture 6 (Mo January 27): 5 (Subgroups) Homework 1 due Homework 2 (sec. 4, 5, 6)

Lecture 7 (We January 29): 5 (Subgroups, continuation) Deadline to drop a course with no permanent record notation

Lecture 8 (Fr January 31): 6 (Cyclic groups)


WEEK 4:

Lecture 9 (Mo February 3): 6 (Cyclic groups, continuation)

Lecture 10 (We February 5): 6 (Cyclic groups, continuation)

Lecture 11 (Fr February 7): 7 (Generating sets of groups) Homework 2 due, Homework 3 (sec. 7,8,9)

WEEK 5:

Lecture 12 (Mo February 10): 8 (Groups of permutations, Cayley's Theorem)

Lecture 13 (We February 12): 8 (Groups of permutations, Cayley's Theorem, continuation)

Lecture 14 (Fr February 14): 9 (Orbits, cycles, alternating groups)

Selected proofs for Midterm 1

WEEK 6:

Lecture 15 (Mo February 17): 10 (Orbits, cycles, alternating groups, continuation) Homework 3 due, Homework 4 (sec. 10,11)

Lecture 16 (We February 19): no lecture, instead Midterm Exam 1 on Lectures 1-13 on Wednesday February 19, 2014, during class time.

Lecture 17 (Fr February 21): Correction of Midterm Exam 1 on the blackboard


WEEK 7:

Lecture 18 (Mo February 24): 10 (Cosets, Lagrange's Theorem)

Lecture 19 (We February 26): 10 (Cosets, Lagrange's Theorem, continuation)

Lecture 20 (Fr February 28): 11 (Direct Products and finitely generated abelian groups)


WEEK 8:

Lecture 21 (Mo March 3): 11 (Direct Products and finitely generated abelian groups, continuation)

Lecture 22 (We March 5): 11 (Fundamental Theorem of Finitely Generated Abelian Groups)

Lecture 23 (Fr March 7): 13 (Homomorphisms) Homework 4 due Homework 5 (sec. 13,14,15)



WEEK 9:

No classes - Spring Break

WEEK 10:

Lecture 24 (Mo March 17): 14 (Quotient groups, Fundamental Homomorphism Theorem)

Lecture 25 (We March 19): 14 (Quotient groups, Fundamental Homomorphism Theorem, continuation)

Lecture 26 (Fr March 21): 15 (Quotient group computations, simple groups) Homework 5 due

Selected proofs for Midterm 2


WEEK 11:

Lecture 27 (Mo March 24): no lecture, instead Midterm Exam 2 on Lectures 13-25 (Monday March 24, 2014) during class time.

Lecture 28 (We March 26): Correction of Midterm Exam 2 on the blackboard.

Lecture 29 (Fr March 28): 34 (Isomorphism Theorems)


WEEK 12:

Lecture 30 (Mo March 31): 18 (Rings and Fields) Homework 6 (sec. 18,19)

Lecture 31 (We April 2): 19 (Integral domains)

Lecture 32 (Fr April 4): 19 (Integral domains, continuation)

WEEK 13:

Lecture 33 (Mo April 7): 20 (Fermat's and Euler's Theorems)Homework 6 due Homework 7 (sec. 20,22,23)

Lecture 34 (We April 9): 20 (Fermat's and Euler's Theorems, continuation)

Lecture 35 (Fr April 11): 21 (The field of quotients of an integral domain)


WEEK 14:

Lecture 36 (Mo April 14): 22 (Rings of polynomials)

Lecture 37 (We April 16): 22 (Rings of polynomials, continuation)

Lecture 38 (Fr April 18): 23 (Factorization of polynomials over a field)


WEEK 15:

Lecture 39 (Mo April 21): 26 (Homomorphisms and quotient rings) Homework 7 due Homework 8 (sec. 23,26)

Lecture 40 (We April 23): 27 (Prime and maximal ideals)

Lecture 41 (Fr April 25): 29 (Extension Fields) Last day of classes. Homework 8 due.