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01/08/15

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ALGEBRA II

Basic Course Information

Classes start September 7 and will be held Tuesdays and Thursdays 11:30am-1:00 pm in MC213 (McLaughlin College). I will largely follow the book by Durbin (see textbook below), where we will concentrate on chapters 10-16. But we will start with an extended review session on monoids, groups and rings, in which we will touch upon some themes that are not covered in Durbin's book, such as free structures. For this part of the course (which will occupy the month of September), you will have to rely on your class notes, although I will provide suggestions for further reading in class.

Evaluation will be based on

- 5 class tests, your 4 best of which will be counted at 10% each toward your final grade (the tests are scheduled for October 5, November 2, November 30, February 1, March 1); [added on Feb 8:] an additional class test will be offered on March 29, so that you have the chance that the best 4 of a total of 6 tests will be counted at 10% each;

- 10 short homework assignments, to be counted at 2% each toward your final grade,

- an individual project (to be assigned in January), with an in-class presentation in March, counting 10% toward your final grade,

- a final exam (TO BE HELD APRIL 11, 9-11am, in VH 1158), counting 30% toward your final grade.

THE FINAL EXAM WILL DRAW PROBLEMS FROM ANY OF THE TOPICS COVERED BY THE COURSE THROUGHOUT THE YEAR. In the final class on April 3, we'll have a review session.

There will be no make-up class tests. If you miss a test, you'll receive 0% on that test, unless you have a medical reason validated by a MD (doctor's note, to be presented within one week after the test missed), in which case the weight of the test missed can be added to the weight of the final on your request. If you choose that option, than your final will count 40% (in case one test was missed), while the tests count 30%, for which the best 3 out of the number of tests written will be chosen. Otherwise the weights for tests and final stay at 40% and 30% respectively, with the 4 best out of the number of tests written being counted.

The University has strict policies regarding academic dishonesty, please see:

http://www.yorku.ca/secretariat/policies/document.php?document=69

 

Office hours (in N605Ross): Tuesdays and Thursdays 2:30-3 pm.

 

Textbook

John R. Durbin, Modern Algebra, An Introduction (5th Edition), Wiley.

Other recommended books:

John B. Fraleigh: A First Course in Abstract Algebra (7th Edition), Addison-Wesley.

David S. Dummit and Richard M. Foote, Abstract Algebra (3rd Edition), Wiley.

Saunders Mac Lane and Garret Birkhoff, Algebra (3rd Edition), AMS Chelsea Publishing.

 

Individual projects

The projects concern working out all details of particular sections of the book by Durbin, including solutions of exercises, and giving a half-hour in-class presentation of the main contents. Projects are listed in order of appearance in class, tentative dates for oral in-class presentation are also listed. Your lecture notes should be discussed with me well before the talk.

Final write-ups of your presentation (incl. solutions of problems) are due April6, the very latest.

RSA algorithm (Section 60): M. Louie (March 8)

Linear codes (62): C. Onilla (March 8)

Standard decoding (63): T. Isho (March  13)

Error probability (64): J. Isaac (March 13)

Lattices (66): A. Razumova (March 15)

Boolean algebras (67): A. Menezes (March 15)

Finite Boolean algebras (68): A. Akhvlediani (Mach 20)

Switching (69): A. Calamici (March 20)

Burnside's Counting Theorem (54): N. Lockshin (March 22)

Crystallographic groups (58): H. Tran (March 22)

The Euclidean group (59): N. Cultraro (March 27)

 

Home work assignments

Assignment 1, due Thursday, Sept 21, in class (strictly). 

Problem 1: For two given elements a, b in a semigroup S, assume ab = ba and prove that for all n = 1, 2,.., one has the law (ab)^n = (a^n)(b^n). Is the assumption ab = ba a necessary condition for the law?

Problem 2: Prove that the multiplicative monoid N of natural numbers 1, 2,... is a free commutative monoid.

Problem 3: Is the multiplicative monoid N isomophic to the additive monoid N_0 x N_0 x ...x N_0 (n times), for any n = 1, 2,...? Prove your claim.

Assignment 2, due Thursday, Oct 5, in class (strictly). 

Problem 1: Let S be a monoid. Find a subgroup G of S with the property that any monoid homomorphism f: H-->S (H any group) has its image in G.

Problem 2: Let X be a set, and let ~ be the least congruence relation on F_Mon(X) with xy ~ yx for all x, y in X. Prove that F_Mon(X)/~ is a free Abelian monoid over X.

Problem 3: Find a monoid S such that there is no group G that contains S as a submonoid.

Assignment 3, due Thursday, Oct 19, in class (strictly). 

Problem 1: Let G be the group of all symmetries of the square (given by rotations and reflections). Establish a multiplication table for that group.

Problem 2: Let G be a group generated by elements a, b with a^4 = e, b^2 = e, baba = e. Show that G has at most 8 elements. Is there such a group with precisely 8 elements?

Problem 3: Let H, K be normal subgroups of a group G whose intersection contains only e_G, and whose union generates the whole group G. Prove that G is isomorphic to  H x K. (Hint: Prove that hk = kh for all h in H and k in K, and consider the map  H x K --> G with (h,k) |--> hk.)

Assignment 4, due Tuesday, Nov 7, in class (strictly).

Problem 1: Let R be a commutative unital ring, and let S be a multiplicative submonoid of R. Define an equivalence relation ~ on R x S by (a,s)~(b,t) if there is r in S with rat = rbs. Let a/s denote the ~-equivalence class of (a,s). Show that with

a/s + b/t = (at+bs)/st  and  (a/s)(b/t) = ab/st

one can make RxS/~ into a commutative, unital ring, and that j(a) = a/1 defines a homomorphism j of unital rings from R into RxS/~ that maps S into the group of invertible elements of RxS/~

Problem 2 (=continuation of Problem 1): Let R and S be as above, and let phi: R --> T be a homomorphism that maps S into the group of invertible elements of the commutative unital ring T. Show that there is unique homomorphism psi: RxS/~ ---> T of unital rings with psi.j = phi.

Problem 3: Let R be the least subring of the complex numbers that contains Z and a root of the polynomial x^2 + 3 = 0. Prove that R is not a unique factorization domain.

Reminder: Test #2 will be held in class on Thursday, November 2nd. Themes: Fundamental Thm on fin. gen. ab. groups, Homomorphism Thm, Correspondence Thm (both for groups and rings), the monoid ring R[M], polynomial rings, ideals (proper, maximal, prime), integral domains, Euclidean domains, PIDs, UFDs.

Mock test questions: 1. Find all quotient groups of Z_8.   2. Let phi: R --> S be a non-zero homomorphism of rings. Prove that phi is injective if R is a field.   3. For commutative and unital rings R and S, let phi: R--> S be a unital ring homomorphism, and let a, b be in S. Show that there is a unique unital ring homom. psi R[x,y] --> S with psi(x) = a, psi(y) = b and psi|R = phi.  Prove that im(psi) is the least unital subring of S containing a, b and im(phi).   4. Problem 37.8 from Durbin's book. [The Problems from Sections 37-41 are all useful practice problems.]

Assignment 5, due Thursday, November 23 (in class).

Problems 1-3: Solve problems 42.5, 42.7, 42.11 of Durbin's book.

Reminder: Test #3 will be written November 30, during class time. The predominant theme will be field extensions, as covered in class since the last test. Sections 42-45 in Durbin's book have most of the material; re-reading Section 32 would also be useful.

Assignment 6, due Tuesday, January 23 (in class).

Problems 1-3: Solve problems 46.13, 46.14 and 46.16 of Durbin's book - after correcting any incorrect claims possibly made in these problems.

Assignment 7, due Thursday, February 8 (in class).

Problems 1-3: Solve problems 47.6, 48.2, and 48.4 of Durbin's book.

Reminder: Test #4 will be written February 1, during class time. The predominant theme will be separable and normal field extensions E over F, how to construct isos in Gal(E/F), and the Fundamental Theorem of Galois Theory. Sections 46-48 in Durbin's book have most of the relevant material and good practice problems.

Assignment 8, due Thursday, February 22 (in class):

Problems 1-3: Solve problems 49.2, 49.7 and 49.11 of Durbin's book.

Assignment 9, due Thursday, March 8 (in class):

Problems 1-3: Solve problems 50.2 (that is: explain thoroughly why the trisection of a 60°angle is equivalent to the constructability of the solution of a certain cubic polynomial equation), 50.9 and 53.10 of Durbin's book.

Reminder: Test #5 will be written March 1, during class time. In addition to Galois theory, you should be familiar with solvability by radicals, solvable groups, ruler-compass constructability, and the basics of group actions.

Assignment 10, due Tuesday, March 27 (in class):

Problems 1-3: Solve problems 53.18, 55.10, 55.14.

Reminder: Test #6 will be written March 29, during class time. Topics not tested in earlier tests include group actions (Section 53 PLUS material covered in class) and the Sylow theorems (Section 55). Furthermore, familiarity with the notions of lattice and Boolean algebra (Sections 66, 67) is assumed.

 

Unofficial Final Grades (last 3 digits of student no; in parentheses: score on final exam):

Student # Grade Student # Grade Student # Grade
086 C (53) 244 D (30)    
973 C+ (43) 225 A+ (95)    
107 D (33)        
325 A (83)        
843 D+ (39.5)        
722 A+ (86)        
576 B (62)        
488 B (70.5)        
402 B (46.5)        

 

 

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