MATH 5402 - Abstract Algebra II

Class Information:

Online class information:

  • A youtube video that I made to show you how to use zoom for our class.

Tests:

Homework:

7.1

1, 2, 3, 11, 14(a,b,c)

 

Solutions are here.
7.2

1

 

Solutions are here.
7.3

1, 5, 6, 8(a,c), 17, 18(a), 24


Solutions for all problems except for #5 are here (note the solution for 5 is incorrect in this link.   See instead the link below).

 

A Solution for 5 is here.

 

Typo: In the solutions for 24(a) in the middle of the page it says that "phi^(-1) (J) is a subgroup of S."   It should say that it is a subgroup of R.
7.4 8, 9, 10, 14(a,b), 15

Solutions are here.

MATH 4560 

Also do these from problems from the Math 4560 website:


HW 1 - 2, 3(a,b), 6, 8, 9

HW 2 - 2, 3

HW 3 - 6

HW 4 - 1, 2, 4, 7

HW 5 - 2, 5, 6, 8, 10

HW 6 - 1, 3, 4, 7

HW 7 - 1 - 7

 
 

 

TEST 1 COVERS EVERYTHING UP TO HERE

 
8.1

3

 

Solutions are here.
8.2

3, 5(a)

 

A solution to #3 is here.   Mimic example 2 on page 273 for 5(a).

 

An easier way to do #3: Let R be a PID and P be a prime ideal of R.  

 

case 1: If P = {0}, then R/{0} is isomorphic to R which is a PID.

 

case 2: If P is not {0}, then by a theorem in class on 2/19, since R is a PID and P is a prime ideal not equal to {0}, we have that P is maximal.   Thus, R/P is a field.   Thus it's only ideals are { 0 + P } and R/P.   These are both principal since { 0 + P } = ( 0 + P ) and R/P = ( 1 + P ).
9.1

4

 

A solution is here.
9.2

3, 6(a,b)

 

The solutions are here.

 

Note on 9.2 #3 solution:
 
At one point in the solutions we have that 
 
f(x) = f(x) g(x) b(x)
 
in F[x] where F is a field.    We know that F[x] is a PID (which includes being an integral domain). 
 
Thus we can change the above equation to this one:
 
f(x) * [  g(x) b(x) - 1 ] = 0
 
Since we are in an integral domain, either f(x) = 0   or    g(x) b(x) - 1 = 0.
 
We don't have that f(x) = 0.    Hence g(x) b(x) = 1.    Therefore, g(x) and b(x) are units in F[x].   The only units in F[x] are the constant polynomials.   Thus g(x) and b(x) are constants in F.
9.4

1(a,b), 2(a,b), 6(a,b,c)

 

Solutions are here.

MATH 4560

Also do these from problems from the Math 4560 website:


HW 8 - 2, 3, 4, 5
13.1

From the book: 1, 2, 3, 4.     Solutions are here.  

From a handout.    Solutions are here.

 

Typos: Let t denote theta.

 

In 13.1 #2, in my above solution I have (a+c)t^3 and it should have been (b+c)t^3.     See this solution instead (thanks to Santiago).

 

In 13.1 #3, in my above solution I have 

t^5 = t ( t^2 + t ) = t^3 + 2t

It should have been t^3 + t^2.  See this solution instead (thanks to Santiago).

 

MATH 4560

Also do these from problems from the Math 4560 website:


HW 8 - 6, 7
13.2

From the book: 2, 3, 14.    Solutions are here.

From a handout.    Solutions are here.
 

 

TEST 2 COVERS EVERYTHING UP TO HERE

 
13.4

1, 2, 4
 

Solutions to 1 and 2 are here.   A solution to 4 is here.
13.6

3

 

A solution is here.
14.1

From the book: 5.     A solution is here.

 

Also do these:

A) Find the Galois group of x^4 - 2 over Q.   Prove that the group is not abelian.

B) Find the Galois group of the polynomials in 13.4 #2 and #4 over the field Q.    

C) Construct the elements of the Galois group of the finite field F_4 over Z_2.    (Hint: First construct F_4 using x^2 + x + 1)

 

A solution to A is here.   Solutions to B and C are here.
14.2

3

 

A solution is here.

    Lecture Notes: