QUALIFYING EXAMS
There are two sets of written exams (Basic and Advanced) which are intended to measure a students overall mastery of standard material. The
exams are administered in August and January, during the week just before
each semester begins. In
Math 611 Midterm, Wednesday 10/22/13, 7PM-8:30PM.
Instructions : Exam time is 90 mins. There are 5 questions each worth 10
points for a total of 50 points. Calculators, notes, and textbooks are not
allowed. Justify all your answers carefully.
1
Q1. Let G
Math 611 Homework 7
Paul Hacking
November 14, 2013
Reading: Dummit and Foote, 12.212.3 (rational canonical form and Jordan normal form), 11.111.3 (review of vector spaces including dual space).
Unfortunately the material on bilinear forms is not covered i
Math 611 Homework 8
Paul Hacking
November 26, 2013
Reading: Dummit and Foote, 10.4, 10.5.
All rings are assumed commutative with 1.
(1) (This question clears up some confusion that arose in HW6, Q9 and
Q10.) Let R be a ring and I and J ideals of R.
(a) Sh
Math 611 Homework 6
Paul Hacking
November 5, 2013
Reading: Dummit and Foote, 10.1-10.3 and 12.1.
All rings are commutative with 1. We say M is a free R-module of rank
n if M
Rn . (We only consider free modules of nite rank.) Justify your
answers carefully
Math 611 Homework 5
Paul Hacking
October 17, 2013
Reading: Dummit and Foote, Chapter 8.
Justify your answers carefully.
(1) Let R be an integral domain and a, b R. Recall that we say a and b
are associates if a divides b and b divides a. Show that the fol
Math 611 Homework 4
Paul Hacking
October 10, 2013
All rings are assumed to be commutative with 1 unless explicitly stated
otherwise.
Reading: Dummit and Foote, Section 6.3 and Chapter 7.
Justify your answers carefully.
(1) Let G = x, y, z | yz 2 xy be the
Math 611 Final, Friday 12/6/13 Friday 12/13/13.
Instructions : This is a take-home exam. Solutions are due by 5PM on Friday
12/13/13. Show all your work and justify your answers carefully.
(1) (a) Let G be a group and x G an element. Dene the centralizer
Math 611 Homework 3
Paul Hacking
October 19, 2013
Reading: Dummit and Foote, 5.5,4.5,3.4.
Justify your answers carefully.
(1) Let
G = O(2) = cfw_A R22 | AAT = I
be the group of 2 2 orthogonal matrices and
H = SO(2) = cfw_A O(2) | det A = 1
the subgroup o
Math 611 Homework 1
Paul Hacking
October 19, 2013
Reading: Dummit and Foote, 1.14.3 (presumably much of this will be
review).
Justify your answers carefully.
(1) Establish the relation bab = a1 between the generators a (rotation by
2/n) and b (a reection)
Math 611 Homework 2
Paul Hacking
September 17, 2013
Reading: Dummit and Foote, 4.4, 4.5, 5.5.
Justify your answers carefully.
(1) (a) Is SL2 (Z/3Z) isomorphic to S4 ?
(b) Is PSL2 (Z/3Z) isomorphic to A4 ?
[Hint: Consider the action on the set P1 (Z/3Z) of