Download Qualifying Examination in Algebra August 2000

Qualifying Examination in Algebra
August 2000
The field of real numbers will be denoted by ~, the field of complex numbers by C and
the field of rational numbers by Q. The ring of integers is denoted by Z.
Note! You must show sufficient work to support your answer. \Vrite. your answers
as legibly as you can on the blank sheets of paper provided. Use only on.e side of
each sheet; start each problem on a new sheet of paper; and be sure to number your
pages. Put your solution to problem 1 first, and then your solution to number 2, etc.
H some problem is incorrect, then give a counterexample.
1. Let 4>: Z E9Z E9Z E9Z -+ Z EB Z EB Z be the group homomorphism which is given
by multiplication
by the matrix
.
4 10 IS 16
lvf=
2
2
6
.
2
[ 6 12 24 18]
Express the cokernel of 4>as a direct sum of cyclic groups. (Recall that the cokernel
eze~ .)
of <pis defined to be . Z
Image...
.
./
.
2. Let R be the quotient ring R = Q[xJ/ (X3 - 3x - 3), let 7t':Q[xJ -+ R be the quotient
map and let x = 7t'(x) be the image of x under the quotient map.
(a:) Show thatR is a field.
(b) Find the dimension of R as a vector space over Q.
(c) express 1/ (1 + x) as a polynomial of degree :5 2 in x.
3. Let G be a finite group of odd order. Show that every subgroup of G of index 3 is
nprmal.
.
4. Let A(t), B(t) E ~[tJ with deg(A(t)) = ~eg(B(t)) = 5. Then show there is a
nonzero polynomial p(x, y) E IR[x,yJ so that p(A(t), B(t)) = O. (That is, p(A(t), B(t))
is the zero polynomial in IR[t].)
5. Let A1nxn(C) be the n x n over the complex numbers. For A E i\1nxn(C) let
C;\ := {B E 1Vlnxn(C) : AB = BA} be the set of elements of Mnxn(C) that commute
with A.
.
(a) Show CAis a subspace of Nlnxu(C).
(b) Show dimCA 2:n.
.
(c) Give an example of a matrix A where dimCA
= n.
6. Let ~r be a finite dimensional vector space and T: V -+ V a linear map. Let
ker(T) be the kernel of T and Image(T) the image of T. If Image(T) ~ Image(T2)
then show V
= Image(T)
E9ker(T).
7. Let I be an ideal in the commutative ring R which is maximal among the nonfinitely generated ideals of R. Pro\'e that I is a prime ideal of R. (The hypothesis on
I says that if J is an ideal of R with I c J and I =1=
J, then J is a finitely generat~d
ideal of R.)
1
8. List 4 proper prime ideals of the polynomial ring Clx, y] which contain the ideal
(4X2 - 13xy + 3y2). Which of your ideals are maximal ideals?
9. Do there exist proper subgroups G1 and G2 of the group (Q, +), with G1 EBG2 isomorphic to (Q, +)?
10. (True or False. If true, prove it. If false, give a counterexample.) If H is a normal
subgroup of the finite group G, then there exists a subgroup of G which is isomorphic
to ~.
... ... .. ..
....--.-....