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

 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:

• Exercises give you a chance to practice and refine your basic calculus skills. They will not be collected.
• Graded homework: There will be seven graded homework due at regularly spaced dates to be annouced in class. The lowest grade will be dropped. Each of the other six is worth 5% of the final garde.

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.