Introduction to Algebra II (MATH 30b, Spring 2007)


[ Policies & Procedures | Schedule of Lectures ]

Day & Time: Monday and Wednesday 3:40 — 4:55pm

Location: Goldsmith 0116

Instructor: Joel Bellaiche

Office Hours: Tuesday 2:30pm-4:30pm and by appointment

Teaching assistant: Alex Charis

Textbook: John B. Fraleigh First Course in Abstract Algebra seventh edition, Adison-Wesley , 2003. ISBN 0201763907.

Prerequisites: Introduction to Algebra I or the equivalent.

Examinations: There will be two in class tests. The first will be on Wednesday, February 13 and the second on Wednesday, April 2. The final exam is a take-home exam


Assessment: The course grades will be computed as follows:
30% Homework
20% First test
20% Second test
30% Take-Home Final exam

Tests and Exams: There are no makeups for missed tests. You must take the final examination at the time scheduled by the university; no final exams will be given earlier.

Homework: There will be two kinds of homework:

Written work: We write to communicate. Please bear this in mind as you complete assignments and take exams. You must explain your work in order to obtain full credit; an assertion is not an answer. student or professor.

Help: Help is available if you have trouble with homework or lecture material. My office hours are a good place to start. Discussions with your classmates can also be very helpful and are strongly encouraged. You may also ask the TA.


Schedule of Lectures

Class Topic Read   Suggested Exercises Due
Jan 16 Introduction. Rings and Fields §18 18:3,7,8,11,15,19,27,33,38,43,46,52,55
Jan 23 Rings and Fields. Domains §18,19 19:3,4,14,15,16,17,23,30
Jan 28 Domains. Little theorem of Fermat §19,20 20:11,12,23,27,28
Jan 30 Euler's theorem. The field of quotients of a domain §20,21 21:4,15
Feb 4 Rings of Polynomials §22 22:7,12,18,23 Problems set 1 Solutions
Feb 6 Rings of Polynomials. Division of polynomials; roots §22,23 23:9,14,25,34,35
Feb 11 Irreducible polynomials. Eisensetein's criterion §23 23:16,17,18,19,20,21,31,36
Feb 13 Midterm I (Solutions) Problems set 2 Solutions
Feb 25 Unique factorization of Polynomials. Kernel of an homormorphism and Ideals §23,26 26:18,30
Mar 3 Factor Ring. Prime and Maximal Ideals. §26,27 27:10,11,13
Mar 5 Prime and Maximal Ideals. Princial Ideals. Principal ideal domains §27 27:15,18,19,28,34,35, Problems set 3
Mar 12 Principal ideal domains. Z, and K[X] (K a field) are principal ideal domains. Application to the minimal polynomial of a matrix. §27
Mar 19 Extensions Field §29 29;1,6,9,19,20,21,25,29,30,31,32 Problems set 3
Mar 24 Vector Spaces. Algebraic Extensions §30,31 30:1,4,5,6,7. 31:1,2,3,12,14,27
Mar 26 Geometric Constructions. §32 32:1,2,4 Problems set 4
Mar 31 Review Training exercises for midterm II 32:1,2,4
Apr 2 Midterm 2
Apr 7 Finite fields 33 33:1,2,3,8,9,10,11,12
Apr 9 Finite Fields. Introduction to Galois Theory 33,48
Apr 14 Automorphisms of fields 48 48:1,2,3,4,9,22,23,25,29
Apr 16 Automorphisms of fields. The isomorphism extension theorem 48,49 49:1,2,3,7 Exercises 5
Apr 28 Separable extensions. 51 Exercises 6
Apr 30 Galois extensions. The main results of Galois theory.