MATH 113 : Introduction to Abstract Algebra (Fall 2009):


Instructor: Alvaro Pelayo, 791 Evans Hall, apelayo@math.berkeley.edu

Lectures: Tuesdays and Thursdays, 8:10-9:30am, Room 4 Evans Hall

Office hours: Tuesdays 11:15-12:30 and Thursdays 12:15-1:30.

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 September 22, 2008 (25%), Midterm Exam 2 on November 3 (25%), cumulative Final Exam (35%) on December 16. 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.

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.

Prerequisites: Math 54 or equivalent knowledge of linear algebra.

Syllabus: (From the course catalog): "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 (Th Aug 27): 0, 1 (Sets, maps, cardinality, equivalence relations) Homework 1 (sec. 0,2,3,4)

Week 2:

Lecture 2 (Tu Sep 1): 2, (Binary operations), 3 (Isomorphism between binary operations)

Lecture 3 (Th Sep 3): 4 (Groups)

Week 3:

Lecture 4 (Tu Sep 8): 5 (Subgroups), 6 (Cyclic groups) Homework 1 due, Homework 2 (sec. 4, 5, 6)

Lecture 5 (Th Sep 10): 6 (Cyclic groups, continuation)

Week 4:

Lecture 6 (Tu Sep 15): 6 (Cyclic groups, continuation)

Lecture 7 (Th Sep 17): 7 (Generating sets of groups)

Homework 2 due Friday September 18, before 5pm, at 791 Evans Hall. You can also turn it in at Thursday's class.

Week 5:

Lecture 8 (Tu Sep 22): no lecture, instead Midterm Exam 1 on Lectures 1-6 (Tuesday September 22, 2008) during class time.

Lecture 9 (Th Sep 24): Correction of Midterm Exam 1 on the blackboard.

Deadline to drop a course without the Dean's approval Friday Sep 25

Week 6:

Lecture 10 (Tu Sep 29): 8 (Groups of permutations, Cayley's Theorem) Homework 3 (sec. 7,8,9)

Lecture 11 (Th Oct 1): 9 (Orbits, cycles, alternating groups)

Week 7:

Lecture 12 (Tu Oct 6): 10 (Orbits, cycles, alternating groups, continuation)

Lecture 13 (Th Oct 8): 10 (Cosets, Lagrange's Theorem) Homework 3 due, Homework 4 (sec. 10,11)

Week 8:

Lecture 14 (Tu Oct 13): 11 (Direct Products and finitely generated abelian groups)

Lecture 15 (Th Oct 15): 11 (Fundamental Theorem of Finitely Generated Abelian Groups), Homework 5 (sec. 13,14,15)

Week 9:

Lecture 16 (Tu Oct 20): 13 (Homomorphisms) Homework 4 due

Lecture 17 (Th Oct 22): 14 (Quotient groups, Fundamental Homomorphism Theorem),

Week 10:

Lecture 18 (Tu Oct 27): 15 (Quotient group computations, simple groups)

Lecture 19 (Th Oct 29): 34 (Isomorphism Theorems), 18 (Rings and Fields) Homework 6 (sec. 18,19)

Homework 5 due Friday October 31, before 5pm, at 791 Evans Hall. You can also turn it in at Thursday's class.

Week 11:

Lecture 20 (Tu Nov 3): no lecture, instead Midterm Exam 2 on Lectures 7, 9-18 (Tuesday November 3, 2008) during class time.

Lecture 21 (Th Nov 5): 18 (Rings and Fields, continuation)

Week 12:

Lecture 22 (Tu Nov 10): 19 (Integral domains)

Lecture 23 (Th Nov 12): 19 (Integral domains, continuation)

Week 13:

Lecture 24 (Tu Nov 17): 20 (Fermat's and Euler's Theorems)

Homework 6 due Friday November 21, before 5pm, at 791 Evans Hall. You can also turn it in at Thursday's class.

Lecture 25 (Th Nov 19): 21 (The field of quotients of an integral domain), 22 (Rings of polynomials) Homework 7 (sec. 20,22,23)

Week 14:

Lecture 26 (Tu Nov 24): 22 (Rings of polynomials, continuation), 23 (Factorization of polynomials over a field)

Thursday November 27 is a holiday

Week 15:

Lecture 27 (Tu Dec 1): 26 (Homomorphisms and quotient rings), 27 (Prime and maximal ideals)

Lecture 28 (Th Dec 3): 29 (Extension Fields) Last day of classes. Homework 7 due

Final Exam December 16.