M901, Exam 1: Wednesday, October 12, 2011
Instructions: Do any three problems. You may assume any problem in doing another, even if you dont do
the problem you assume, as long as you dont use A to do B and B to do A, or A to do B and B to do C
and C to do
M901, Assignment 7: Due Monday, November 21, 2011
Instructions: Do any three problems.
Background: A eld k is said to be perfect if either char(k ) = 0, or char(k ) = p > 0 and every element of k
is the pth power of some element of k (i.e., k p = k , wher
Background:
I: There are three basic methods for coming up with normal subgroups of nite groups.
(1) You can apply the Sylow theorems. For example, the Sylow theorems tell us immediately that a group
of order 111 must have a normal subgroup of order 37.
(
M901, Assignment 5: Due Friday, October 28, 2011
Instructions: Do any three problems.
Background: Let N and H be groups and let : H Aut(N ) be a homomorphism. We dene N H to be the set
N H with the composition law (n1 , h1 )(n2 , h2 ) = (n1 (h1 )(n2 ), h1
M901, Assignment 4: Due Friday, October 7, 2011
Instructions: Do any three problems.
Background: two categories C and D are said to be isomorphic if there exist functors F : C D and
G : D C which are mutually inverse to each other (i.e., the compositions
M901, Assignment 3: Due Friday, September 23, 2011
Instructions: Do any three problems.
First we establish some background.
Let S be the category of sets.
Let V be the category of nite dimensional real vector spaces. Let IdV be the identity functor from V
M901, Assignment 2: Due Friday, September 16, 2011
First we establish some notation.
Given m Z, where Z is the group of integers, let f : Z Z/mZ be the quotient homomorphism. Denote
f (i) by [i]m .
Let G be a group. A commutator of G is an element of G of
M901, Final Exam: Thursday, December 15, 2011
Instructions: Do any three problems. You may cite without proof facts proved on the homeworks or previous
exams, or other problems on this exam (but you are not allowed to cite Problem m while solving Problem
M901, Exam 2: Friday, December 2, 2011
Instructions: Do any three problems.
(1) Prove that S4 is solvable but not nilpotent.
(2) Let Q = cfw_1, i, j, k be the group of quaternion units (i.e., the group of order 8 where 1 are
in the center, (1)2 = 1, i2 =
M901, Practice Problems, Field Theory
(1) Let k K be a normal algebraic extension of elds. Let E be the subset of K of all elements of K
separable over k . Show that E is a normal extension of k .
(2) Give an example of a nite extension k K of elds and an