Homework assignments for Math 28b, Spring 1998


The book used for the course was A Book of Abstract Algebra, by Charles C. Pinter (McGraw-Hill, 1990). We covered Chapters 17-29.

Problem Set 1. Due Thursday, Jan. 29, 1998

  1. Page 175: B1,B4 (see page 170 of the book for the definition of F(R).)
  2. Page 179: K1,K2,K4.
  3. Let B consist of all matrices where a and b are real numbers. Is B a subring of M2(R)? Is B a field? Explain.
  4. Let R be a ring with unity, and let R* denote the set of invertible elements. Show that R* is a group, where the group operation is the multiplication in the ring. (You may want to refer to problem I.1-5 on page 178.)
  5. Let B consist of all matrices where a,b,c,d,e,f are in Z5, Show that B is a subring of M3( Z5). How many elements are there in B*? Write down a subgroup of B* of order 125.

Problem Set 2. Due Thursday February 5, 1998

  1. Page 185 A1,A6.
  2. Pages 185-186 B1-B4.
  3. Suppose f:A -> B is a ring homomorphism, where A and B are rings with unity. Show that f(0) = 0. Is it true that f(1) = 1? Prove or give a counterexample.
  4. Page 187 E1,E7.
  5. Page 188 G2, H4.

Problem Set 3. Due February 12, 1998

  1. Page 196 B1-B4
  2. Pages 196-197 C1; D1,D2
  3. Consider the ring Z[x] of polynomials with integer coefficients.
    1. Show that the quotient ring Z[x]/< x > is isomorphic to Z. (Suggestion: consider the map F: Z[x] -> Z given by F(p) = p(0). What is its kernel?)
    2. Try a similar argument for Z[x]/< x-1 >.
  4. Pages 198-199 H1,H2,H4

Problem Set 4. Due Wednesday February 25, 1998

This problem set is a little longer than most, but you have a few extra days to work on it.
  1. Page 205 A1,A2
  2. Page 205 C1,C2,C3
  3. Page 214 B1,C3
  4. Pages 215-216 F1-F5. (Think about F6, but you don't have to do it.)

Problem Set 5. Due Thursday March 12, 1998

  1. Page 223 B1,B3, C1,C2
  2. Page 224 D1,D2
  3. Page 225 G1-G3, G5
  4. Using the Euclidean Algorithm find the gcd of 49,349 and 15,555.
  5. Suppose that a and n are relatively prime integers. Show that a has an inverse in Z/n. Show that for any b, the congruence ax = b has a solution in Z/n.
  6. Use the Euclidean algorithm to find numbers s and t with 96s + 37t = 1. Explain how this finds the inverse of 37 (mod 96) and the inverse of 96 (mod 37).

Problem Set 6. Due Thursday March 19, 1998

  1. Page 234 A2,A4
  2. Page 235 D1
  3. Page 236 E1-E6,E8(a)
  4. Page 237 G1-G4

Problem Set 7. Due Thursday April 2, 1998

  1. Page 247 A2,A7
  2. Page 249 F1-F3
  3. Page 255 A1,A4,A5
  4. Page 257 F1,F3,F4
  5. Show that gcd(x3+1,x2+1) = 1 in Q[x], by exhibiting two polynomials r(x) and s(x) with (x3+1)r(x) +(x2+1)s(x)=1. (Hint: use the division algorithm, as we did for finding gcd's in Z.)

Problem Set 8. Due Thursday April 9, 1998

  1. Page 265 A1,A2,B5
  2. Page 266 D1,D2,D3
  3. Page 267 F1,F2,F3
  4. Page 269 I1,I3

Problem Set 9. Due Thursday April 23, 1998

  1. Page 276 A1
  2. Page Page 277 C1-C5
  3. Page Page 278 E1-E6

Last HW! Problem Set 10. Due Thursday April 30, 1998

  1. Page 289 A1,A4.
  2. (a) Consider R as a vector space over Q. Is it finite dimensional? Justify your answer. (b) Same question, for C as a vector space over R.
  3. Page 297 A1-A3.