Abstract Algebra Homework Problems

Solutions to past homeworks (click here)

Due 3/24/10

I. Read Section 50

II. Do 50 #2, 4, 6, 7, 8, 10, 22, 23, 21 (I think these will be easier in this order)

Due 3/10/10

I. Read Section 49

II. Do 49#1, 2, 4, 5, 7

Due 3/3/10

I. Read Section 48

II. Do 48 #1, 2, 4, 5, 7, 10, 15, 17, 18, 20, 32, 35, 39

Due 2/24/2010:

Part A

I. Do 31# 32, 33

Part B

I. Read Section 32

II. Do Section 32 #3, 4, 5, 6, 8

Part C

I. Read Section 33

II. Do Section 33 #1, 2, 3, 9, 11, 12

Due 2/10/2010:

Part A

I. Section 29 #29, 30, 34

II. Read Section 30

III. Do 30 #4, 5, 9, 16, 26

Part B

I. Read Section 31

II. Do 31 #3, 4, 5, 22, 23,

Due 2/3/2010:

Part A

I. Do Section 27 #1, 5, 24, 30

II. Let N₁ and N₂ be two prime ideals of the commutative ring R. Show that the intersection of N₁ and N₂ is not necessarily a prime ideal.

Part B

I. Read Section 29

II. Do 29# 2, 4, 6, 12, 14, 16

Due 1/27/10

I. Do Section 26#26, 27

II. Read Section 27

III. Do Section 27#31

IV. Do these:

a) Let R be the real numbers. Compute (and simplify) (x² -1)² in R[x]/<2x^2-4x+6>.

b) Show that for any commutative ring R, R[x]/<x> is isomorphic to R.

c) Consider the set N={2p(x)+(x-1)q(x) | p(x), q(x) in Z[x]}. Show that N is an ideal of Z[x] but not a principal ideal.

Due 1/20/2010

Part A

I. Read Section 23

II. Do Section 23 #1, 3, 12, 14, 34, 36

Part B

I. Read Section 26

II. Do Section 26 #4, 8, 9, 18, 22, 24