Math 6121, Abstract Algebra I

Fall 2005

• Instructor: Matt Baker
• Time and place: MWF 12:05 P.M. - 12:55 P.M, Skiles 149
• E-mail: mbaker@math.gatech.edu
• Office: Skiles 263
• Office Hours: Monday 2-3, Wednesday 11-12

If you haven't already done so, please fill out the Online Course Survey.

Exam schedule:

• The take-home final exam is due on Wednesday, December 14.

Twelfth homework assignment:

• Read Sections 13.1-13.2 of the textbook.
• Section 13.1 #2,5
• Section 13.2 #2,4,5,7,10,14,19,20

Eleventh homework assignment:

• Read Sections 8.3,9.1-9.5 of the textbook.
• Section 8.3 #5 (see also pp. 229-230)
• Section 9.1 #6,8,14
• Section 9.2 #10
• Section 9.3 #1,3
• Section 9.4 #12,13,14

Tenth homework assignment:

• Read Sections 7.5,7.6,8,1,8.2 of the textbook.
• Section 7.6 #3,7
• Section 8.1 #3,6,7,10
• Section 8.2 #1,3,5,6

Ninth homework assignment:

• Read Section 7.4 of the textbook.
• Section 7.3 #24,25,29,30,31
• Section 7.4 #6,8,9,11,13,15,27

Eighth homework assignment:

• Read Sections 7.2 and 7.3 of the textbook.
• Section 7.1 #26,27
• Section 7.2 #1,3,4,10,12
• Section 7.3 #6,7,8

Seventh homework assignment:

• Read Sections 5.5 and 7.1 of the textbook.
• Section 5.5 #6,8,9
• Section 7.1 #1,7,8,15,16,21,25

Sixth homework assignment:

• Read Sections 5.1,5.2 of the textbook.
• Section 5.1 #14,15,17ab,18
• Section 5.2 #2ab,3ab,4ab,5,6,14

Fifth homework assignment:

• Read Sections 3.4, 3.5, and 4.6 of the textbook.
• Section 3.3 #7
• Section 3.5 #3,4,9
• Section 4.3 #11,13,29
• Section 4.5 #16,18,22
• Section 4.6 #2,4

Fourth homework assignment:

• Read Sections 2.3, 4.3, and 4.5 of the textbook.
• Section 2.3 #16,26
• Section 4.3 #5,8,23,24,25,26,30
• Section 4.5 #8,12

Third homework assignment:

• Read Sections 2.4,2.5,3.3 and 4.2 of the textbook.
• Section 2.1 #6
• Section 2.2 #6,7
• Section 2.3 #21,22,23
• Section 2.4 #8,14,15
• Section 3.1 #35
• Section 3.2 #5,9

Second homework assignment:

• Read Sections 2.1,2.2,3.1,3.2, and 4.1 of the textbook.
• Section 3.1 #1,3,4,5,9,10,12,14,24,25,36,40,41
• Section 3.2 #5
• Section 4.1 #4
• Bonus problem: In how many ways can one 5-color the vertices of a regular pentagon that is free to move in (a) two dimensions? (b) three dimensions?

First homework assignment:

• Read Chapters 0 and 1 of the textbook.
• Section 1.1 #9,10,24,26
• Section 1.2 #10,18
• Section 1.3 #2,11,15
• Section 1.4 #10
• Section 1.6 #9,14,17
• Section 1.7 #14,16,23

Useful links:

• Polya's enumeration method
• Handout on Cauchy's theorem
• Handout on Sylow's theorems
• Handout on Finitely generated abelian groups
• Handout on the classification of groups of order p^3
• Handout on Eisenstein's criterion

Course outline:

This graduate-level course in abstract algebra is the first in a two-course sequence which also includes Math 6122. Topics to be covered will include:

• Groups:

  subgroups, homomorphisms, quotient groups, Lagrange's theorem, permutation groups,group actions, the class equation, the Sylow theorems, simple groups and composition series, free groups, direct and semidirect products, the structure theorem for finitely generated abelian groups.

• Rings:

  ideals, quotient rings, Chinese remainder theorem, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings over fields.

• Vector spaces:

  linear transformations and matrices, bases and linear independence, dimension, tensor products, dual spaces, multilinear algebra.

• Modules:

  submodules, quotient modules, free modules, finitely generated modules over a principal ideal domain, rational and Jordan canonical forms via module theory.

• Fields:

  algebraic and transcendental extensions, splitting fields, algebraic closure, arithmetic of finite fields, separable and inseparable extensions, classical straightedge and compass constructions.

Prerequisites:

Math 4107 and one of Math 2406, Math 4305, or permission of instructor.

Exams:

There will be 2 in-class midterm exams during the course of the semester, plus a cumulative take-home final exam at the end of the course.

Homework:

Homework will be assigned on a regular basis.

Grading Policy:

The two midterm exams will each count for 25% of your grade, the final will count 40%, and homework will count 10%.

Collaboration:

On the homework sets, collaboration is both allowed and encouraged. However, you must write up yourself and understand your own homework solutions. Any academic dishonesty on the midterm or final exams, if detected, will result in a score of zero for that exam.