Basic Problems: Don't hand these problems in. You should do them before the Core Problems in the same section, to help your understanding.
Core Problems: Everyone must turn these problems in. Always justify your answer, even if the question does not explicitly say so.
Advanced Problems: Students registered for Math 6000 must turn these in, and Math 4000 students hoping to get an A should do a reasonable number of these.
Basic Problems: 1. Sec. 1.1 # 2
Core Problems: 2. Sec. 1.1, #4 bdgh (be sure to use induction), 6, 8a
Basic Problems: Sec. 1.2 # 1, 3, 6
Core Problems: Sec. 1.1 # 7 (use complete induction); Sec. 1.2 # 2, 4, 5, 7
Extra Credit: Sec. 1.2 # 10
Basic Problems: Sec. 1.2 # 6, Sec. 1.3 # 4
Core Problems: Sec. 1.2 # 8, 9, 13, 14 Sec. 1.3 # 12
Advanced Problems: Sec. 1.2 # 20 (you may assume gcd(a,b) = 1)
Basic Problems: Sec. 1.3 # 5, 20 bcde, 21 ab
Core Problems: Sec. 1.3 # 11 (Hint: use a. in doing b.), 14, 19, 20g, 21c
Also do the following problem: (a) What is the last digit of 19 to the 62 power? What is the last digit of 19 to the 23 power?
(b) What are the last 2 digits of 7 to the 62 power? What are the last 2 digits of 7 to the 23 power?
(Hint for the last problem: You should not have to compute any very high powers. Use modular arithmetic and simplify as you go along.)
Advanced Problems: Sec. 1.3 # 16 (Hint: work out some examples and try to see what the pattern is), 29.
Basic Problems: Sec. 1.4 #1 (omit Z mod 12), 3,4
Core Problems: Sec. 1.4 # 5, 7, 13.
Also do the following problems.
1. Find the multiplicative inverse of 26 in Z mod 105, or explain why it does not exist. Do the same thing for 45 in Z mod 105.
2. (a) Show that if R is any ring, then in the addition table for R, every element appears exactly once in each row. Use commutativity of addition to deduce the same fact about columns. (Hint: Write down an equation which is equivalent to the stated problem. Notice that "exactly once" amounts to an existence and a uniqueness statement.)
(b) There exists a ring R with four elements {0,1,a,b} where 0 is the additive identity, 1 is the multiplicative identity, and 1+1 = a. Using (a) work out the addition and multiplication tables for R; include brief explanations of how you got each entry.
Advanced Problem: Sec. 1.4 #12.
Basic Problems: Sec. 2.3 # 3, 4, 8
Core Problems: Sec. 2.3 # 2, 6de, 9b. Note: 13 is postponed until next week.
Basic Problems: Sec. 2.4 # 1bcd, 3
Core Problems: Sec. 2.3 # 9cg, 13, 15 [Hint: Draw a picture of the roots of the LHS and use # 4], 21
Advanced Problems: Sec. 2.3 # 18 [Give an explicit of the units, not just a condition they must satisfy. You don't have to find the field of quotients.]
Basic Problems: Sec. 3.1 # 1 be
Core Problems: Sec. 2.4 # 1 e, 2 ac, 8, 9. (For #8, try to do as much of the problem as possible without computing alpha explicitly. For #9, your tables must involve only the 4 elements listed in the statement of the problem.
Sec. 3.1 # 1 d, 5 (Here, f(x) and g(x) are in R[x] for some ring R.)
Basic Problems: Sec. 3.1 # 2 a, 10 abc
Core Problems: Sec. 3.1 # 2 cd, 10 def, 11, 13, 14 bd.
Note: In class, I made 13 a bonus.
Hints: On 10 and 13, Cor. 1.5 is helpful.
On 14, apply a sequence of ring operations starting from c (for example, taking powers of c, taking rational multiples of c, addition and subtraction) until you get 0. This sequence of operations can be used to figure out a polynomial in Q[x] that has c as a root.
Have a good break!
Basic Problems: Sec. 3.2 # 1, 5c
Core Problems: Sec. 3.2 # 2 a, 3 ace, 6 c, 11
Advanced Problem: Sec. 3.2 # 15
Basic Problems: Sec. 3.3 # 3, 4 ab
Core Problems: Sec. 3.3 # 2 adfj, 4 c, 5, 6 a, 7, 8
Hint for #7: Look at Example 7 (c) on p. 110 of the book. To understand the pattern, you may want to see what happens for some small values of p.
Advanced Problem: Sec. 3.3 # 10
Note: In class, I made the advanced problem a bonus.
Basic Problems: Sec. 4.1 # 1, 2, 3
Core Problems: Sec. 4.1 # 4 bcdf, 5, 7, 8, 15 abcd
Advanced Problem: Sec. 4.1 # 4 g (Hint: This is easier if you use Proposition 3.1 of this chapter. I will allow you to use this, even though we haven't gotten there yet.)
Core Problems: Sec. 4.1 # 6, 9, 14 ac, 16, 17
Note: # 9 and 14 should be easy after class on Tuesday, so if you cannot do them now, wait until after class.
Basic Problems: Sec. 4.2 # 1
Core Problems: Sec. 4.2 # 3 ace, 4b, 12, 13 abc, 16 ac
Advanced Problems: Sec. 4.2 # 13 de Note: On # 3, use the fundamental homomorphism theorem, as we did in an example in class. I'll do some more examples on Tuesday.