Homework Assignments
Math 449 - Modern Algebra
Spring Term 2003
Dr. Monks
Due dates for these assignments will be announced in class or via email.
Assignment #1 (Due Feb 11, 2003)
Read the course syllabus.
Do the following review exercise (the file is in LaTeX format so you can edit it directly in Scientific
Workplace in the math lab)
Assignment #2 (Due Feb 13, 2003)
Section 8.1 #1c,2,4,6a,7,8,12,14,17,22,27b,28
Assignment #3 (Due Feb 18, 2003)
Section 8.2 #2,3b,f,d,h,4,5b,d,6b,d,7d,8,12,16,17a,18,19
Assignment #4 (Due Feb 20, 2003)
1. Let p(n) be the partition function. Compute p(n) for
1<=n<=8. (Do NOT try to use the Hardy-Ramanunjam formula!!)
2. Classify all finite Abelian groups of order 256.
3. Derive a recursive formula for p(n). (Hint:
Define p(n,k) to be the number of partiions of n having maximum value
k,
derive a recursive formula for p(n,k)
and use it to get a formula for p(n). Alternatively you can define P(n,k) to be
the
number of partitions of n
having k terms and use the same approach to get a formula for p(n).)
Assignment #5 (Due, Feb 25, 2003)
Section 8.3 #4,6,8,11,12,14,16,18,19,23,24a
Assignment #6 (Due, Feb 27, 2003)
Answer the questions on this assignment (click
here).
Assignment #7 (Due Mar 4, 2003)
Section 8.4: #1b,c (Note: Use cycle notation),2,6,7,8,10,12,16
Assignment #8 (Due Mar 6, 2003)
Let Q be the quaternion group defined on page172. Note: in
the following problems use the notation 1,i,j,k,-1,-j,-k wherever possible,
don't make the multipication table or list subgroups or conjugacy classes using
matrix notation.
1. Make the multiplication table for Q. (see page 55 for hints)
2. Find Z(Q).
3. Compute Q/Z(Q) and construct its multiplication table.
4. Classify Q/Z(Q).
5. Find all of the subgroups of Q.
6. For each subgroup of Q, determine if it is normal in Q or not.
7. For each normal subgroup H of Q, compute Q/H and classify this group
(you do not have to make the multiplication table for each quotient group).
8. Find all of the conjugacy classes of Q.
Assignment #9 (Due Mar 18, 2003)
Section 8.5: #2, 3, 8, 14
Assignment #10 (Due Mar 18, 2003)
Section 9.1: #1,2,9,11,12,13,14,20,22,23
Assignment #11 (Due Mar 20, 2003)
Section 9.2: #3a,4,9,10,16,17,18
Assignment #12 (Due Mar 25)
#1. Is Z[sqrt(-2)] a PID? Is it a Euclidean Domain? Prove or
disprove. (Do this question last.)
Section 9.3: #4,6,7,8,9,11,13,14,19a,b,c,21
Assignment #13 (Due Mar 25)
Section 9.4: #1,2,4,6,7,8,12
Assignment #14 (Due Mar 27)
Section 9.5: #1,2,4,5,6,7,8,12
Assignment #15 (Due Apr 1)
Section 10.1: #2,3,4,5,21b,c
Assignment #16 (Due Apr 1)
Section 10.1: #6,14,16,18,26,30,31
Assignment #17 (Due Apr 3)
Section 10.2: #1,2,3,4,5,6,7,8,9,10,11,12,13,14,18
Assignment #18 (Due Apr 8)
Section 10.3: #3d,4,6,8,10,11,12,16
Assignment #19 (Due Apr 10)
Section 10.4: #1,2,5,6,12
Assignment #20 (Due Apr 15)
Section 10.4: #8,9,10,13,16
Assignment #21 (Due Apr 15)
Section 10.5: #4,5,6,7,8,10,12b,15
Assignment #22 (Due Apr 22)
Section 10.6: #2,4,6,10,11,12,13,15a (make the multiplication table for
it as well),21
Assignment #23 (Due Apr 22)
Construct a splitting field for x4+x3+1
over Z2, and verify that it is a splitting field.
Assignment #24 (Due Apr 24)
Section 11.1: 2,6,7b,8,10b,13
Assignment #25 (Due Apr 29)
Section 11.2: 1,2,4,6,7a,8,9
Assignment #26 (Due May 1)
Construct the complete Galois correspondence diagram for the extension Q(21/4,i)
over Q as discussed in problem #11, page 386. [Hint: D4 has
five subgoups of order 2 and three subgroups of order 4.] Give a basis for each of the
intermediate fields in your diagram.
Assignment #27 (Due May 6)
Section 11.3: 1b (show its radical!),5,6,10,17a,b,c,d,18
Assignment #28 (Due May 8)
Construct the subgroup diagram for the quaternion group described in exercise 14 on page
172.
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