Math 5126
Wednesday, March 28
Seventh Homework Solutions
1. Section 14.4, Exercise 6 on page 596. Prove that F p (x, y)/F p (x p , y p ) is not a simple
extension by explicitly exhibiting an innite number of intermediate subelds.
Set K = F p (x p , y p )
Math 5126
Friday, January 31
First Homework Solutions
1. Since R/I is a Noetherian ring, it is Noetherian as an R/I -module. The submodules
of R/I considered as an R-module are precisely the same as the submodules of R/I
considered as an R/I -module; thus
Math 5126
Wednesday, February 5
Third Homework
Due 9:05 a.m., Wednesday, February 12
1. Let R be a PID, let M be a nitely generated R-module, and let N be an R-submodule
of M . If N R as R-modules, prove that M N X as R-modules for some R-module
=
=
X . I
Math 5126
Monday, February 17
Third Homework Solutions
1. By the structure theorem for nitely generated modules over a PID, M Rn T ,
=
where T is a torsion module. Since N is not a torsion module, we see that M = T and
hence n = 0. We deduce that M R X fo
Math 5126
Wednesday, January 22
First Homework
Due 9:05 a.m., Wednesday January 29
1. Let R be a commutative ring and let I be an ideal of R such that R/I is a Noetherian
ring. Suppose there are only nitely many ideals of R contained in I . Prove that R i
Math 5126
Wednesday, January 29
Second Homework
Due 9:05 a.m., Wednesday, February 5
1. Let K be a eld, let R = K [x, y] (the polynomial ring in two variables over K ), and let
I = (x, y), an ideal of R.
(a) Prove that if M and N are nonzero R-submodules
Math 5126
Wednesday, February 12
Fourth Homework
Due 9:05 a.m., Friday, February 21
1. Section 12.3, Exercise 23 on page 501.
(3 points)
2. Section 12.3, Exercise 25 on page 501. (Note that n is an eigenvalue, and that similar
matrices have the same rank.
Math 5126
Monday, March 3
Fifth Homework
Due 9:05 a.m., Monday, March 17
1. Section 13.4, Exercise 3 on page 545.
(3 points)
2. Let p be a prime, let F be a subeld of the eld K of characteristic p, and let a K
n
be algebraic over F . Prove that a p is sep
Math 5126
Wednesday, March 26
Sixth Homework Solutions
1. Section 14.2, Exercise 2 on page 581.
for the element 1 + 3 2 + 3 4.
Determine the minimal polynomial over Q
Let denote the above element. We nd a Galois extension K containing , and then
consider
Math 5126
Monday, March 24
Seventh Homework
Due 9:05 a.m., Monday, March 31
1. Let K be a Galois extension of Q with degree 12 and Galois group G.
(a) Prove that the number of subelds of K of degree 4 over Q is 1 or 4.
(b) If G does not have a normal subg
Math 5126
Friday, February 7
Second Homework Solutions
1. (a) If M N = 0, then there exist 0 = m M and 0 = n N . We now have nm M
because M is an R-submodule, and mn N because N is an R-submodule. Since
M N = 0, we deduce that mn = 0, which contradicts th
Math 5126
Monday, February 10
Sample First Test no. 1. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let k be a eld, let V be a vector space over k, and let (e1 , e2 , e3 , e4 ) be a k-basis for
V . Dene a linear transformation T : V
Math 5126
Wednesday, March 19
Fifth Homework Solutions
1. Section 13.4, Exercise 3 on page 545. Determine the splitting eld and its degree over
Q for x4 + x2 + 1.
4 + x2 + 1 = (x2 + x + 1)(x2 x + 1), so the roots are 1 3 . Therefore the
x
2
splitting eld
Math 5126
Wednesday, February 26
Fourth Homework Solutions
1. Section 12.3, Exercise 23 on page 501. Suppose A is a 2 2 matrix with entries
from Q for which A3 = I but A = I , where I is the identity matrix. Write A in rational
canonical form and in Jorda
Math 5126
Wednesday, April 25
Tenth Homework Solutions
1. Set
A=
a1
,
0a
N=
01
.
00
Then A = aI + N , N 2 = 0 and we see that
Ar = ar I + ar1 rN .
Therefore the (1,2) entry of Ar is rar1 , and is in particular nonzero for r > 0. We
conclude that A has inn
Math 5126
Wednesday, April 18
Ninth Homework Solutions
1. 18.1.2 on page 852. Let : G GLn (F ) be a matrix representation. Prove that the
map g det( (g) is a degree 1 representation.
Obviously the image of the map is in GL1 (F ) (that is the nonzero eleme
Math 5126
Monday, April 30
Solutions to Sample Final Problems
1. Let : G GL2 (C) denote the natural inclusion. Then this is a matrix representation,
and we would like to show that it is irreducible. If this is not the case, then it contains a
one dimensio
Math 5126
Monday, January 16
First Homework
Due 9:05 a.m., Monday January 23
1. Let R be a commutative ring and let I be an ideal of R such that R/I is a Noetherian
ring. Suppose there are only nitely many ideals of R contained in I . Prove that R is
a No
Math 5126
Monday, January 23
Second Homework
Due 9:05 a.m., Monday January 30
1. Let R be an integral domain and let M be a Noetherian R-module.
(a) Prove that M has a maximal free submodule (i.e. M has a free submodule F such
that if E is a submodule of
Math 5126
Monday, January 16
Policy Sheet
Course
Prereq
Instructor
Book
Ofce
Telephone
E-mail
Room
Math 5126, Abstract Algebra CRN (index number) 14415
Math 5125 (Abstract Algebra)
Peter A. Linnell
Abstract Algebra 3rd. ed. by David S. Dummit and
Richard
Math 5126
Friday, February 10
First Test Review
The test will cover chapter 12, and sections 13.1,2. Topics will include
1. Noetherian rings and modules
2. Torsion submodule
3. Fundamental structure theorem for nitely generated modules over a PID.
4. Stru
Math 5126
Monday, April 2
Second Test Review
The test will cover most of sections 13.4 14.6. Topics will include
1. Splitting elds
2. Separable polynomials and derivative
3. The Galois group of a separable polynomial of degree n is isomorphic to a subgrou
Math 5126
Wednesday, April 25
Sample Final Problems since Second Test
1. Let G be a nite subgroup of GL2 (C) which contains the matrices
01
,
10
Prove that
10
.
0 1
| tr(g)|2 = |G|. Hint: show that the corresponding representation on C2
gG
is irreducible
Math 5126
Tuesday, January 24
First Homework Solutions
1. Since R/I is a Noetherian ring, it is Noetherian as an R/I -module. The submodules
of R/I considered as an R-module are precisely the same as the submodules of R/I
considered as an R/I -module; thu
Math 5126
Wednesday, February 1
Second Homework Solutions
1. (a) Since 0 is a free submodule, it is immediate from the Noetherian property that M
has maximal free submodules.
(b) If M is not a torsion module, then there exists m M such that rm = 0 for all
Math 5126
Friday, February 10
Third Homework Solutions
1. Section 12.2, Exercise 4 on page 488. Prove that two 3 3 matrices are similar if
and only if they have the same characteristic and same minimal polynomials. Give an
explicit counterexample to this
Math 5126
Wednesday, February 29
Fourth Homework Solutions
1. Section 13.2, Exercise 2 on page 529 (part of). Let g(x) = x2 + x 1 and let h(x) =
x3 x + 1. Obtain a eld with 8 elements by adjoining a root of f (x) to the eld F
where f (x) = g(x) or h(x) an
Math 5126
Wednesday, March 14
Fifth Homework Solutions
1. Let f denote the minimal polynomial of a over K and let n denote the largest nonnegn
n
ative integer such that f is a polynomial in x p . Then we may write f = g(x p ) for
some polynomial g whose d
Math 5126
Wednesday, March 21
Sixth Homework Solutions
1. Section 14.2, Exercise 14 on page 582. Show that Q( 2 + 2) is a cyclic quartic
eld; i.e., is a Galois extension of degree 4 with cyclic Galois group.
Set = 2 + 2 and K = Q( ). Since satises x4 4x2