Class Information:
Online class information:
 A youtube video that I made to show you how to use zoom for our class.
 How to use TinyScanner: https://www.youtube.com/watch?v=quEH1AvDhAQ
 How to use CamScanner: https://www.youtube.com/watch?v=84SOQ3EY5bQ
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
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. 
Also do these from problems from the Math 4560 website:
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. 
Also do these from problems from the Math 4560 website:


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).

Also do these from problems from the Math 4560 website:


13.2 
From the book: 2, 3, 14. Solutions are here. 
TEST 2 COVERS EVERYTHING UP TO HERE 

13.4 
1, 2, 4 
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)

14.2 
3
A solution is here. 
Lecture Notes:
 Wednesday, 1/22
 Monday, 1/27
 Wednesday, 1/29
(Note: We stated on 1/29 that R/I is a ring when I is an ideal of R. We didn't
prove it that day. But we did prove it on 2/5 below.)  Monday, 2/3
 Wednesday, 2/5
 Monday, 2/10
 Wednesday, 2/12
 Monday, 2/17
 Wednesday, 2/19
 Monday, 2/24
 Wednesday, 2/26
 Monday, 3/2
 Wednesday, 3/4
 Monday, 3/9
 Wednesday, 3/11  Test 1
 Monday, 3/16
 Wednesday, 3/18  no class this day
 Monday, 3/23
 Wednesday, 3/25
 Monday, 3/30  SPRING BREAK
 Wednesday, 4/1  SPRING BREAK
 Monday, 4/6
 Wednesday, 4/8
 Monday, 4/13
 Wednesday, 4/15
 Monday, 4/20
 Wednesday, 4/22  Test 2
 Monday, 4/27
 Wednesday, 4/29
 Monday, 5/4
 Wednesday, 5/6