## 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.