Math 4340

HW1) D&F p.22 #8,9,18,22,26,27 p.27 #1,2 p.32 #1 p.39 #1,2 due Friday, Feb 3 in class.

HW2) D&F p.40 #13,14 p.44 #1,2,4,5-8 p.85 #1 due Friday, Feb.10 in class.HW3) D&F p.52 #5a,b,6 (hint - 6a, 2nd half - let H consist of 2 transpositions in S_3)

p.60 #2,8,9,11 p.101 #2,3,7,8 due Friday, Feb. 17 in class.

HW4) D&F p.60 #1,13,18 p.101 #2,3,7,8 due Friday, Feb. 24 in class. (The last 4 problems were

assigned last week so this means you have an extension till next week to turn those in. Reason: we

didn't get far enough in class. Also it means you will have a much shorter assignment than usual which

fits well with the break next Monday & Tuesday.)

HW5) D&F p.28 #10-13 p.65 #2,7,8 p.86 #18 p.111 #8,9,11 (in #8 show only that A_4 has no subgroup of order 6;

in #11 consider the elements of order 4 and their squares) due Friday, March 3 in class.

HW6) D&F p.45 #21-23 p.122 #7-9 (in #8 show that [G:K] divides n!; use this in #9) p.146 #5,6,7

due Friday, March 10 in class.

NOTE: This week Beihui is changing her office hours to 2-4PM Tuesday (the 14th) because of the prelim on Wednesday.

PRELIM in class on Wed. March 15 (closed book). POSTPONED till Friday, 17th March because of likely Cornell

closure due to winter storm.

HW7) D&F p.146 # 8,13 p.106 # 6,7,9,10 (you may use the results of other exercises on this page without proof)

p.230 # 1,4 p.247 # 2,5,6,8 (all rings are assumed to have multiplicative identities which are preserved

by ring homomorphisms; all subrings have the same mult. id. as the parent ring; answers may differ from

D&F which doesn't have these conventions.) due in class, Friday, March 24.

HW8) D&F p.277 #2-6, p.301 #1-5 due in class, Friday, March 31

(Euclidean domains will be covered in class on Monday. Wait until then to do the hw or read enough about the topic

on your own to get started earlier.)

HW9) D&F p.278 #10,11 p.282 #1,2,3,5,6 due in class, Friday, April 14.

Note: As announced in the first week of class, and mentioned several times since, the course material is largely what

is covered in class. The textbook (Dummit & Foote) and the homework are supplementary. A primary purpose of

the course is to teach a new way of thinking: how to use abstract concepts and the axiomatic method to study a wide

variety of mathematical objects. This is very hard to learn well entirely from books.

HW10) D&F p.278 #9,12 p.292 #2,8a,8c(1st part) p.529 #1,2 due in class, Friday, April 21.

HW11) D&F p.344 #7,8,9,18,19 p.356 #3 p.165 #1a,b,2a,b,3a,b due in class, Friday, April 28.

To get started you will need the definition of a R-module M where R is a commutative ring. Think

of M as a 'vector space ' over R (i.e. the scalars are from R and the 'vectors' form an abelian gtoup M)

and you won't go wrong. See D&F for the formal definition. A 'cyclic module' is one which is generated

by a single element (analogous to a cyclic group or a 1-dimensional vector space). The Fundamental

Theorem of Abelian Groups, needed for the problems on p.165, will be covered next week along with

modules.

HW12) D&F p.166 4a,b,c p.356 2,4,5,7 p.469 10,11,12 p.488 3,4 p.499 #4 (need not find P) .

(p.469 #12 and p.499 #4 were added on Friday, May 5.) Due in class, Wed., May 10.

Note: The Final Exam will be on Thursday, May 18 7:00 PM - 9:PM in MLT406: Malott Hall 406.

It will be closed book and cumulative, with about one-third of the questions on group theory and the

rest on material covered since.

Office Hours (Exam week): I will have office hours on Tuesday, May 16, 11a.m.-12noon in Malott 505.

Beihui's office hours: Wednesday, May 17, 2 - 4p.m. in Malott 218.

Solutions

HW1 Solutions

HW2 Solutions

HW3 Solutions

HW4 Solutions

HW5 Solutions

HW6 Solutions

HW7 Solutions

HW8 Solutions

HW9 Solutions

HW10 Solutions

HW11 Solutions

HW12 Solutions

Text: Dummit and Foote, 3rd. ed., John Wiley & Sons

Exams: Final, Prelim in class about half way through, possible 2nd Prelim (take-home)

near the end of the term.

HW: Approximately weekly (perhaps 11 or12 assignments). You may collaborate with a couple of other

students from the class but are encouraged to try every problem yourself first. If you do collaborate

state the names of the other students. The grading won't depend on whether you collaborate, but make

sure to write the solutions in your own words.

Grading policy: The final grade will depend approximately 15% on the hw and the remaining 85% on the

exams (whether 2 or 3).

Instructor: (Prof.) Shankar Sen

Office Hours: MWF 11.10 - 11.50, Malott 505 (and by appointment)

TA: Beihui Yuan

email: by238@cornell.edu

Office Hours: Thursdays, 2-4 PM, Malott 218