Math 103b Winter 2008
Homework Assignments
Homework 1
Due Friday, January 11
Chapter 12: #2, 6, 18, 22, 23, 26, 40, 42, 46
Homework 2
Due Friday, January 18
Chapter 13: #6, 13, 14, 16, 22, 24, 30, 41, 54
Chapter 14: #4, 10, 12, 18
Comments: For 13.24, you may use without proof that the square root of d is irrational if d is not a perfect square.
For 13.41(a,b), you should assume that p is prime (this is an error in the statement of the problem.)
In several problems, it may be helpful to use the binomial theorem. If you do not remember it, wikipedia has a good review of it (search for "binomial theorem".)
In 13.22, let me clarify the definition of the ring in question. R is the set of all functions from the set of real numbers to itself. Two such functions f and g are added using the following formula: [f+g](x) = f(x) + g(x). Similarly, we multiply two functions by defining [fg](x) = f(x)g(x). This makes the set of all such functions into a ring. Make sure you understand what the 0 element and the 1 element are in this ring.
In 13.54, recall the proof of Fermat's little theorem from last quarter. The argument used there should give you some ideas about how to approach this one.
Homework 3
Due Friday, January 25
Chapter 14: #6, 31 (see comment below!), 33, 34, 36, 56
Chapter 15: #11, 14, 26, 36, 40, 50
Comments and hints:
14.31---I want you to do more than the problem asks. I want you to show that the ring in question is isomorphic to Z
n
for some n. Follow the method presented in class.
14.36---Be careful, this one is somewhat different from the example in 14.31 and the similar example presented in class.
15.36---Show that any such homomorphism must send 1 to 1, and that the homomorphism is completely determined by where 1 goes.
15.40---Think about the kernel of the homomorphism.
Homework 4
Due Friday, February 8
Chapter 16: #12, 28, 31, 40, 48
Chapter 17: #2, 6, 8, 10(b) 12, 14
In addition, determine if the following polynomials are irreducible over Q: 5x
3
+ 7x + 2, 21x
3
- 6x
2
+ 7x -2 (Consider the rational-root test.)
Comments and hints:
16.31---Hint: Use Fermat's little theorem. Don't try to multiply out the right hand side.
16.40---Hint: Use the same method as in Example 3 on page 298 (which was also explained in class.)
17.8---Use the result of exercise 17.6.
Homework 5
Due Friday, February 22 (note the new due date! Also, one new problem has been added, #11.)
Chapter 18: #2, 4, 10, 11, 13, 14, 17, 18, 22, 23, 30
Comments and hints:
18.10---mimic the proof of Theorem 17.5 in the text.
18.18, 18.22---Use the methods of Examples 1 and 2 in the text (also done in class.)
Homework 6
Due Friday, March 7 (I added more problems to the Homework on Friday 2/29).
Chapter 31: #6, 8, 10, 17(a), 24
Chapter 23: #3, 4, 5, 6, 7, 8.
Comments and hints:
31.#6 "binary vectors" means that it is a code over Z
2
, the only kind we considered in class.
31.#8. From the statement of Theorem 31.2, it is not clear that the code with bigger weight can do anything better, but it can. Look at the proof of Theorem 31.2---the proof shows something a bit stronger than the actual statement of the theorem suggests.
Chap 23---beware that all of these problems are a bit interdependent. You can use in your proofs all the other results we did in class.
Homework 7
Due Friday, March 14.
Chapter 19: #16, 22.
Chapter 20: #1, 6, 17.
Chapter 23: #9, 10, 14, 15, 16.
Comments and hints:
20.#1. Think of this field as a subfield of the real numbers, and apply the theorem we proved in class.
23. #9, 10. These are supposed to be short: just show that 15 degrees is constructible but 40 degrees is not. You can use the fact that 20 degrees is not constructible, which is in the text and which we will do in class.
Department of Mathematics
Math 103A